Season the chicken breasts on both sides with salt and pepper.
Heat 1 tablespoon of the oil in a large nonstick skillet over medium high heat.
Place 4 pieces of the chicken in the skillet and sear until until golden-brown, about 1-3 minutes on each side. Transfer the chicken to a plate and repeat searing with the remaining chicken pieces. Set aside.
Heat 1 tablespoon of oil in the empty skillet.
Add the mushrooms and stir occasionally until most of the moisture has evaporated from the pan, about 8 minutes.
Transfer the mushrooms to a plate and set aside.
Heat the remaining 1 tablespoon of oil in the pan.
Add the onions and bell peppers and season with salt and pepper. Stir occasionally until the onions and peppers have softened, about 2 minutes.
Add in the garlic and rosemary and stir until the garlic is fragrant.
Add in the tomatoes, red wine, and broth and stir until the sauce has thickened slightly.
Add the mushrooms and chicken back into the sauce and bring to a low simmer.
Cover and let cook until the chicken registers 165°F (74°C), about 5-8 minutes.