Season chicken with salt and pepper to taste.
Heat in large nonstick skillet 1 tablespoon of the olive oil. Add chicken to skillet in single layer and cook over medium heat for 4 to 5 minutes per side or until chicken is no longer pink in center. Remove from skillet to plate and cover to keep warm.
Wipe out skillet and then heat in skillet over medium-high heat remaining 1 tablespoon olive oil. Add and cook, stirring over medium-high heat until softened, about 5 minutes mushrooms and red pepper. Add and cook, stirring for 1 minute without browning garlic, 1/2 teaspoon salt and 1/8 teaspoon ground black pepper.
Return the chicken breasts and any accumulated juices from the plate to the skillet, arranging in a single layer. Top each chicken breasts with a slice of cheese.
Cover the skillet and lower the heat to medium-low. Continue to cook until the cheese has melted over the chicken, about 1 to 3 minutes. To serve, spoon the mushroom mixture evenly over chicken. Makes 4 servings.