Cut a 2-inch pocket in the thickest part of each chicken breast.
Dividing evenly, stuff the pockets with the cheese and season with ½ teaspoon salt and ¼ teaspoon pepper.
Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Cook the chicken until golden brown and cooked through, 6 to 8 minutes per side.
Meanwhile, arrange the tomatoes and peppers on a plate. Drizzle with the remaining oil and season with ½ teaspoon salt and ¼ teaspoon pepper. Serve with the chicken.