Heat 1 tablespoon oil in a nonstick skillet over medium-high heat. Add green bell pepper and onion; cook and stir until tender, 8 to 10 minutes.
Heat remaining 1 tablespoon oil in another nonstick skillet over medium-high heat. Add chicken; cook and stir until chicken is crumbly and no longer pink, 8 to 10 minutes. Stir in green bell pepper, onion, salt, and black pepper. Top with cheese slices and cook until melted, 2 to 3 minutes.