Heat 1 tablespoon vegetable oil in large skillet over medium-high heat. Add chicken to skillet, season with salt and pepper and cook until no longer pink, about 4 minutes. Transfer to a clean bowl.
Add the remaining tablespoon of oil and cook 30 seconds. Add vegetables to skillet. Cook until lightly browned, about 3 minutes.
Add chicken back to skillet, leaving any accumulated liquid in bowl behind. Add Kung Pao sauce and peanuts to the skillet, cook until sauce is thickened, about 1 minute. Remove from heat and serve.