Rinse the chicken and pat dry with paper towels. Cut the chicken breasts into thin strips. (Partially freeze chicken breasts first to make this easier).
Combine the sauce ingredients in a small bowl and set aside. Whisk the cornstarch into the egg whites. Coat the chicken strips in the cornstarch mixture. Heat 1 cup oil in the wok. Cook the chicken strips briefly in the hot oil until they turn white. Remove the chicken and drain on paper towels. Clean out the wok and add 1 tablespoon oil. When the oil is hot, add the carrots. Stir-fry briefly and add the peppers.
Make a "well" in the middle of the wok and add the sauce. Heat briefly, then mix the sauce in with the vegetables.
Return the chicken to the wok.
Stir-fry 1 to 2 more minutes. Stir in the green onions. Serve with rice.