Stir the cornstarch, broth, soy, ginger and sesame oil in a medium bowl. Set the mixture aside.
Heat 1 tablespoon oil in a 12-inch skillet over medium-high heat. Add the chicken and stir-fry until it's browned. Remove the chicken with a slotted spoon and set it aside.
Reduce the heat to medium and add the remaining oil. Add the broccoli, peppers and garlic. Stir-fry until the vegetables are tender-crisp. Stir the cornstarch mixture and stir it into the skillet. Cook and stir until the mixture boils and thickens. Return the chicken to the skillet and add the peanuts and cook until the mixture is hot and bubbling. Serve over the rice.