Warm 2 tablespoons of olive oil in a Dutch oven or a sturdy, deep pot over medium-high heat. Season the chicken thighs on both sides with kosher salt and black pepper.
Once the oil is shimmering, add the chicken in a single layer. Sear the chicken for 3 to 4 minutes without moving to achieve a golden crust. Flip and cook for an additional 2 minutes on the other side, then transfer the chicken to a plate. It doesn’t need to be fully cooked at this stage.
Lower the heat to medium, adding the remaining tablespoon of olive oil. Introduce the chopped onion and bell pepper to the pot, cooking until they start to soften, about 4 minutes.
Add the minced garlic, stirring for 1 minute until aromatic.
Pour in the white wine, stirring to loosen any browned bits from the bottom of the pot. Allow the wine to simmer and reduce by half, approximately 3 minutes.
Stir in the crushed tomatoes with juices, sliced mushrooms, Italian seasoning, and balsamic vinegar. Bring the mixture to a gentle simmer, allowing the sauce to cook uncovered for 10 minutes to thicken.
Nestle the partially cooked chicken thighs into the simmering tomato sauce, adding any accumulated juices from the plate. Ensure the chicken is mostly submerged.
Partially cover the pot and let simmer for 15 to 20 minutes, until the chicken is cooked through, the sauce thickens, and mushrooms are tender.
For serving, either shred or dice the chicken if desired. Return it to the sauce to warm through, or leave the thighs whole. Adjust seasoning with additional salt and pepper as preferred.
Serve hot over pasta, rice, or vegetables, topped generously with sauce, fresh parsley or basil, and grated Parmesan.