Stir the cornstarch, broth, soy sauce, ginger and sesame oil, if desired, in a medium bowl until the mixture is smooth.
Heat 1 tablespoon vegetable oil in a 12-inch skillet over medium-high heat. Add the chicken and stir-fry until well browned, stirring often. Remove the chicken from the skillet.
Reduce the heat to medium. Heat the remaining vegetable oil in the skillet. Add the broccoli, peppers and garlic and stir-fry until the vegetables are tender-crisp.
Stir the cornstarch mixture in the skillet. Cook and stir until the mixture boils and thickens. Return the chicken to the skillet. Stir in the peanuts and cook until the mixture is hot and bubbling. Serve the chicken mixture with the rice.