Upon review, I'd simmer it for a bit (15 min) after adding the tomatoes and before adding butter, just to cook down the tomatoes a bit. Be warned, I tend to like my tomatoes less cooked than many people. -I do not simmer sauce all day, for example.
And, @Thanks for the Crepes is correct, this is supposed to be a rustic hunter's dish nothing fancy. Before reading this thread, I had only ever seen it made with bone-in, skin-on chicken -a whole one with all the parts. (well, that and the time I saw it made from rabbit)
HERE's a link to some Italian TV personalities making it. As I watch them, I am struck by how the thigh, viewed from the end appears to be some sort of roll. Are you sure what you had wasn't just a bone-in thigh?
Honestly, doing the whole rolling procedure will not add any flavor to the dish. It will just result in really uniform pieces of chicken. And, by not using the skin (and to a lesser extent the bone) you will lose the gelatin which would normally be in the sauce, giving it a rich, thick, silky mouth-feel. I'd try Crepes' recipe first.
HERE's the restaurant's Yelp page, which has another image of this dish -this time with a lot more mushrooms. Still difficult to see the chicken, though.
The main reason restaurants don't give out recipes is because people don't realize how different restaurant cooking is compared to home cooking. They par-cook then chill dozens of ingredients along with prepping raw foods (like mushrooms) -all in anticipation of order which may or may not come. You can wind up making one plate of lasagna, or 56 plates of it. -And they all have to be identical. (customer service and your account both demand this) So, for a typical restaurant, they have some chicken marked off on the grill but mostly raw, several tubs of sauces simmering in a warmer, a big tub of chilled caramelized onions, and a mini salad-bar's worth of diced and sliced raw items. (plus spices, jarred condiments, pickles, etc.) Most of this stuff can be used tomorrow if no one orders it today.
Anyway, your 'recipe' can't really be real because it would take them more than an hour to serve someone after an order is placed. So, they are using shortcuts. They probably pre-cook mushrooms and have them waiting chilled in their own juice. Same with caramelized onions. Since they toss red sauce on almost everything, there's probably a giant pot of it simmering away on the stove or in a warmer. This works great when managing 300 covers a night, not so well in a home kitchen where you are serving a MUCH more limited menu for any given meal. I suspect they just use rolled meat because it probably appears in several other dishes they serve, not because it improves anything. And, this dish was never really designed as a restaurant dish it's more of a homemaker dish or hearkens back to the days before restaurants when an innkeeper's wife might share the family dinner with people paying for rooms.