Would you be interested?

[Metcarfre]

As a side note, I feel limited menus only work well if they are mixed up often (with a few retained favorites), and supplemented with specials. Again, seasonally available food can play in to this well.

[plummeting_sloth]

I am a sucker for "This is what we made today, eat it or don't" restaurants (the Faculty Dining room is this way) but Met's right... gotta make sure you've got a good rotation/variation.

[Hiramas]

In a medieval setting, it would fit into the theme if you have menus per day.
On mondays you have a pot roast, thursdays a stew, wednesday is pie-day and so on...
Grilled meat like spareribs or grilled chicken can be on every day.

Still, though, side dishes like potatoes and salads may not be very medieval, but people want them. Grilled eggplants are a common and popular vegetarian dish, but not really medieval.
Every restaurant I know that does this skips the authentic menu part at least here.

[plummeting_sloth]

Heck, with a bit of research you could tie them into the bevy of holy days that were always going on.

[Dutch guy]

I would be interested (though I'm probably a bit outside the target area, geographically)

I have a favorite restaurant in the Czech republic that is very similar to what you are proposing which I've visited with friends pretty much every time I've been there. It's located in the brewery house of an old mansion/castle with a nice stone vaulted ceiling on the lower floor and open wooden beam construction on the upper floor. Rough cut wooden tables, big fireplace, etc.
The staff is loud and just on the friendly side of condecending. (Last time we were there the round of drinks ordering started with a simple: "Right you lot! If you want beer, raise your hand and shout Pivo!" Shouted with enough force to get the attention of the entire table of 26)
During the evening there is live entertainment in the form of music, dancing, mock exorcism, mock hanging (with the hanged miraculously coming back to live in the arms of a female visitor), fighting with axes and swords (Yes, really! in the middle of a group of dining people. The fight was well rehearsed and there were support cast watching that no-one got in the way). Above all the food was a somewhat limited choice but very well made. High quality meat with enough sides to give a proper meal. Even for the bigger eaters among us.
They have a website with some foto's here: http://www.krcmadetenice.cz/en/