Close by, you could try one of the 3 Darios in Panzano though my partner and I didn't find the food at his casual restaurant that good. Different perhaps (too much raw veg). Hopefully his formal restaurant is better. FWIW, none of his meat (or at least beef) is from Italy: It's all raised in Spain to his specifications.

The following two places have lots of atmosphere:

Also tried La Castellana just outside Montefioralle which was very nice. Would have loved to have stayed for more courses but we were pretty full from lunch at the following place:

Had lunch at Il Pozzo, a nice, rustic restaurant in the tiny walled village of Monteriggioni and the food was wonderful. The proprietor takes pride of the food, wine and service (even does the final seasoning on the Chianina steaks). A bit further afield but I would not hesitate ti recommend making this track.

We stayed a night at Fattoria Lischeto Agrotourismo near Volterrano which is getting further afield for you. The farm offers meals to large (tour) groups which may not be what you want. They tried to cancel dinner on us (we had requested it months ahead, even reconfirmed a few days before) and they tried to foist us off on one of the local restaurants. We insisted and they brought in a pinch cook who made us a wonderfully delicious dinner, including the best of 4 wild boar pastas my partner had (they used braised meat instead of ground which is what most recipes call for and restaurants use).