Session 1 - Welcome to Gabadona
Thoughts

An interesting experiment, very out of what I have been doing for a while, so I felt the rust in my bones as I went. You can read the rules I am using for this here, if interested.
I had no modules to lean onto, the characters and NPCs which I had asked the players were finished either the night before of the game or a few hours before the session, and the actual structure of the start of the session was sketched out about half an hour before the game itself.

Because of that, I think I was caught a bit flat footed. It has been a while since I had to improvise NPCs and situations like this, I was cracking my brain not to make it feel boardgamey or railroady, and predictably it was to mixed results.
The actual week-to-week structure with actions the players must take was interesting and good, a bit boardgamey but not terrible. The big issue I have to work through is having more events. The core idea is for it to be turn-taking, but I noticed that, aside from Mann who locked into beetle wrestling (to my endless delight), Cellar and Eterna were putting effort into relating to the world and grasping onto something. Eterna in particular ended up not being very sure what to do, as the core actions related to meeting people and getting into fights. I think the random events should be where these hooks are deployed. I had a 1 in 6 chance for them to fire off but that might be too low.
The other thing which is still up in the air is in the actions department. I had it so that each player had 1 action per week, weekend, and holiday, but this might be a bit too slow. At the same time, we don't want to go too fast. Still, I'm leaning towards removing the extra action per weekend and instead just saying that certain week-actions take place in the weekend.
My overall attitude is one of optimism however. The fighting system I cobbled together from Cepheus and Retro Sci-Fi Rules proved to be fast and interesting (contributing to Mann's addiction to beetle wrestling), the characters are very very interesting and the NPCs my players handed to me are also super interesting, and I think the universe is settling into something more solid in my head. It will require fine tuning, but I trust my players and the sturdy system framework I'm running with to see us through.
For the future, I need to be less shy about barefacedly introducing plots and having them develop with open-ended situations. The repeating framework should be generating these circumstances, but it's not a replacement for them. I will cut myself some slack for this session, however, as much of it was planned really fast and out of nowhere.
I also got more ideas for events - thrusting players into situations, developing the NPCs more, having repercussions for those events overflow into the rest of the structure, and having timed projects to guide the actions should all help.
At least with the last point I already have some idea of how to do it; there are seasonal festivals and events which will give the players things to prepare (like rehearsing your Mardi Gras parade and things like that).
It appears this idea does have legs. Who knew!
Recap

2nd Week of Early Autumn, Year 680
We started out with a bit of background on where we are and how we got here: in the southern shore of the Kingdom of Arles, 7 generations after the mysterious vanishing of King Arthur, in the Duchy of Trusca, which is on the border of the Truscan Empire (though not a part of it), in the County of Gabadona, where lies the Gabadona Academy. For generations now, parents have sent their young knightly children to the academy to earn prestige and learn Chivalry.
Our players are:
Alessa Di Ventadorno, played by Mr. Mann. A bookish and responsible young woman with a penchant for theology.
Cora Constanzo, played by Eterna. A meek but brave young woman interested in folklore, though hateful of fairies.
And Fiorello d'Agosto, played by Cellar Gelatin. A calm and modest young man interested in machines, though with a poetic side.
They are joined by two NPCs they were asked to create before the session: the concerned Francisca da Biama and the catty Corina Constanzo, Cora's evil twin and changeling.
After dealing with the paperwork of arrival, they find their way to the hall for the feasting and are introduced to their mentor for the year, Don Antamonte Proscatti, a proud and retired knight who will guide them.
Over the feast, they are introduced to other NPCs. Alessa has a polite conversation with a particularly restrained young lady named Belina di Carim; Fiorello - who wanted to talk to Alessa - finds himself mostly trying to tap into someone else's conversation; and Cora blushes as Rina Di Boiardi, the second year student sitting next to her, puts her arm around her shoulders and tells her she should be making memories.
After the feast they are escorted to the beach for hazing, but Cora manages to sneak out behind Rina. The hazing ritual is to army crawl through garbage and sand towards a party happening on the other side; the winner will gain a pretty sword and be known as the Young Knight of the Year.
Fiorello wants to cheat, so Alessa (unintentionally) uses her book to draw attention to herself as he takes a few steps on his knees to gain an advantage, and wins the race. The veteran who was presiding over this, the 3rd year Carno Carsi - a huge, almost 7 feet tall young man with a powerful body and a mysterious past - congratulates him and gifts him the sword.
Over the party, Fiorello ditched his friend Francisca to hang out with the massive Carno, Cora was herself ditched by the very popular Rina and hung out with Francisca, and Alessa was helped up from the ground by someone named simply the Mourning Dove; a vagrant, poetic teenager who roams the rooftops of the town and waves pigeon feathers on his straw-coloured hair. He mysteriously vanished after this, and so Alessa spent the rest of the night apologising to all for her behaviour.
After this we dropped into the turn structure.
Early Autumn, Week 3
Alessa saw that I had made a Beetle Wrestling minigame and decided to get some beetles. She got 3 - all stag beetles - and named them Cora, Corina, and Rina.
Fiorello started hanging around with Carno and sparred with him, but lost, as he was a much better fighter.
Cora spent a while playing wargames with Belina to get to know her a bit better
Early Autumn, Weekend of Week 3
Alessa decided to create a new area in the game: the Beetle Arena, in the shape of a huge rhino beetle where people put their beetles to fight. She pitted her team against Carno, betting £100, and won! But instead of taking one of Carno's beetles, as is usual, she instead let him keep it.
Fiorello decided to challenge Rina to a duel. Rina was another of the NPCs they made before the game, she is Fiorello's slightly older cousin with a strong dislike for both him and his father. As such, she accepted the duel on the condition they used real swords rather than scabbarded ones. Fiorello innocently accepted. They faced one another at the beach, dramatically, and in a single exchange, Fiorello lost. He had his abdomen cut deeply and had to be rushed to the hospital, the doctor said it would take him until spring to fully heal.
Cora ends up fistfighting Corina after a disagreement over if Fiorello is a scoundrel or not, in front of Rina. She also lost after a single exchange and became estranged from Rina's circle.
Early Autumn, Week 4
Alessa went to catch more beetles and found one.
Fiorello hung out at the Sierra Francatta with Carno's group and one day came upon Carno trying to become closer to the Mourning Dove, and failing. Fiorello approached, trying to patch things up, but Carno said it was alright. There stood in the air the implication that the two knew something about one another.
Cora mostly hung out at the track and field, and had a generally uneventful week.
Early Autumn, Weekend of Week 4
Alessa and Fiorello went after the Mourning Dove and invited him to play boardgames, to get to know him a little bit better, but with not much success. He stuck to being mysterious and a bit eerie.
Cora decided to invite Belina to the amusement park, but she was unavailable, so she decided to go sightseeing instead. She found a little family of tourists from the Truscan Empire discussing the history of the place, how Lancelot had been one of the first masters of the academy and shaped much of its Chivalric bent, how rumours exist that King Arthur used the hoard of a dragon to found it and how that hoard or that dragon might still be around somewhere in the Mountains d'Àrden. They were being guided by an older relative, Namöder von Baldwin, who noticed Cora following them and decided to include her too. Unfortunately, her shifty nature won out and she dashed away from the Baldwins rather than joining them for dinner.
With this, the month ended, and the session with it.