Heat 1 tablespoon of oil in a nonstick skillet over medium-high heat. Add the eggs, swirling to coat the pan in a thin layer. Cook, without stirring, until the eggs are set, approximately 2 minutes. Using a spatula, transfer the eggs to a plate. Finely chop them and set aside.
Heat the remaining 2 tablespoons of oil in the same skillet. Add the carved chicken breast strips. Cook, stirring occasionally, until slightly browned on both sides, about 1 minute. Add the onion to the skillet. Cook until soft, stirring occasionally for approximately 3 minutes. Stir in the vegetables, hoisin sauce, garlic, and soy sauce. Continue cooking with an occasional stir until the vegetables begin to soften, about 5 minutes more.
Add the contents of the yellow rice mix and 2 cups of water to the skillet, and stir to combine. Bring to a boil, then reduce heat to medium-low. Simmer, covered, until the rice is tender and the water is absorbed, approximately 25 minutes. Remove the skillet from the heat and gently stir the reserved eggs into the rice.