Combine sauce ingredients in a small bowl and set aside.

In a large nonstick skillet or wok, heat oil over medium high heat. Add diced onion, carrot, and white parts of green onion to the pan, stirring. Add chicken and sliced garlic, and cook for 3-4 minutes, stirring frequently to avoid burning the garlic. Add the rice to the pan and stir to coat the rice with oil. Push all the rice and other ingredients to one side of the pan and pour the eggs onto the exposed side of the pan. Cook for 2-3 minutes until egg is cooked through. Using a wooden spoon, break up the cooked egg into bite sized bits and stir all the rice with the egg. Add the kimchi and stir fry sauce and cook until liquid has absorbed. Season to taste with salt and pepper. Serve topped with sliced tops of green onions and a fried egg if desired.