In a Dutch oven or equivalent deep, sturdy pot, heat 2 tablespoons olive oil over medium-high heat. Season the chicken on both sides with salt and pepper.
Add the chicken in a single layer after the oil is heated and shimmering (but not smoking).
Cook for 3 to 4 minutes on the first side, until browned, turning the chicken as little as possible to provide a beautiful sear. Flip and brown the other side for about 2 minutes. Put the chicken on a wide, clean dish. The chicken does not need to be fully cooked as it will finish cooking in the sauce.
Turn the heat down to medium. Add 1 tablespoon olive oil, onion, and bell pepper. Cook for 4 minutes, or until the veggies soften.
Add the garlic and cook for 1 minute, or until fragrant.
Pour in the chicken broth. Allow to boil for 3 minutes, or when the broth has been reduced by half.
Stir in the tomatoes with their juices, mushrooms, Italian seasoning, and balsamic vinegar. Bring to a boil. Bring to a boil, then simmer uncovered for 10 minutes, or until the sauce thickens slightly and becomes rich and flavorful.
Place the chicken thighs in the tomato sauce, along with any drippings from the dish. The chicken should be largely immersed, with some visible through the water.
Cook for another 15 to 20 minutes, or until the chicken is cooked through, the sauce has thickened, and the mushrooms are soft.
Remove the chicken to a chopping board if you want to dice or shred it. When it’s cold enough to handle, dice or shred it before returning it to the pot. The chicken thighs can alternatively be served whole.
Season the sauce with extra salt and pepper to taste. Return the chopped or shredded chicken to the sauce and mix to incorporate and reheat through. Alternatively, serve the chicken chunks whole with plenty of sauce on top.
Serve hot over pasta, rice, polenta, or vegetarian noodles, topped with a generous tablespoon of sauce and a sprinkling of parsley and Parmesan.