In a large wok or nonstick skillet, heat half of the oil over medium to medium-high heat. When a drop of water sizzles in the oil, the heat is sufficient. If the oil begins to smoke, it is too hot. Add the garlic and the chicken pieces. Use two wooden spoons or wooden spatulas to stir-fry the chicken and coat them with the hot oil in the pan. Let the chicken cook, stirring with your spatulas frequently for 3 minutes, until the chicken is mostly white.
Sprinkle the chicken with salt and pepper, then add the honey and soy sauce. Stir all together with the spatulas. Let the ingredients cook, stirring every 1-2 minutes, until the chicken pieces get a brown crust on at least one of their sides (5-7 minutes). Once the chicken begins to crust, make sure you watch carefully so they do not burn. When the chicken has a nice brown crust, and the juices that run from them when pierced run clear, remove the chicken to a plate and set aside. Take the wok or skillet off the heat and wipe out with a paper towel. Be careful!
Place wok or skillet back on the heat (medium to medium-high) and heat the other half of the oil. Once water sizzles in the oil, add the bell peppers, onion, and carrot pieces. Mix well with the spatulas, then let them cook, stirring frequently, until they are tender, about 5 minutes.
Stir the chicken pieces into the pan with the vegetables and mix well. Combine the cornstarch and cold water in a small bowl, and mix it well with a fork. Stir the cornstarch mixture into the wok or skillet, and let all ingredients cook for 2 minutes, stirring constantly, until thickened.
Serve immediately over white or fried rice.