Prepare the vegetables: bell peppers should be cored and sliced. Slice the red onion. Clean and halve or quarter mushrooms. Roughly chop garlic. Set aside.
Using paper towels, pat the chicken dry and season with salt on both sides.
Lightly dredge chicken in the flour. Gently shake off excess flour from the chicken pieces and set aside.
Preheat the pan over medium heat. Add olive oil to a large 5-7 quart braiser or Dutch oven with a lid and heat until oil is glistening.
Brown both sides of each piece of chicken until it's golden. Set aside. (The chicken will complete the cooking in the braising process.)
Using the same pan that the chicken was browned in, add vegetables and garlic, and a little more olive if needed.
Sauté vegetables for about 2-3 minutes until they turn bright in color.
Add white wine and deglaze any bits from the chicken and vegetables—about 1 minute.
Add chicken broth, tomatoes, tomato paste, olives, Italian seasoning, dried basil, bay leaf, fresh rosemary, red pepper flakes, remaining salt and pepper. Stir mixture until combined.
Nestle chicken pieces into the vegetables and broth.
Place the lid on the braiser or Dutch oven and reduce the heat to low. Simmer for 30 minutes.
Check after 30 minutes to ensure the chicken and vegetables are tender and cooked through.
Serve with your choice of side and some crusty bread.
