Add one tablespoon of butter and one tablespoon of olive oil to a large, non stick skillet over medium heat. Add the chopped onion and bell peppers and cook until starting to soften and brown, about 6 - 8 minutes. Remove the peppers and onion from the pan to a clean bowl.
Add the remaining tablespoon of butter to the pan over medium heat and cook the mushrooms until browned, about 10 minutes.
Remove the mushrooms from the pan. Add the remaining tablespoon of oil to the pan over medium heat. Add the chicken breasts and brown each side of the chicken, about 3-4 minutes per side.
Add the garlic to the pan with the chicken and cook until fragrant, about one minute.
Stir in the cooked peppers, onions and mushroom. Stir in the crushed tomatoes, 1 teaspoon of salt and black pepper. Cover the pan and simmer until the chicken is fully cooked and is at least 165 degrees F.
Stir in the capers. Taste and add more salt if needed. Serve over pasta.