Begin by seasoning the chicken strips with salt and pepper. In a large skillet or wok, heat 1 tablespoon of sesame oil over medium heat.
Pour the beaten eggs into the pan, allowing them to cook undisturbed for 2-3 minutes until fully set. Gently remove the eggs, roll them into a log shape on a cutting board, and slice them thinly. Set aside.
Increase the heat to medium-high and add another tablespoon of sesame oil to the skillet. Sear the seasoned chicken, stirring frequently, until browned and fully cooked, approximately 3-5 minutes. Remove the chicken and set aside.
Add the remaining tablespoon of sesame oil to the skillet. Introduce the shredded green cabbage, red cabbage, and mushrooms. Sauté the vegetables, stirring occasionally, until they are tender yet crisp, about 3-5 minutes.
In a small bowl, prepare the sauce by whisking together the hoisin sauce, soy sauce, Chinkiang vinegar, garlic paste, ginger paste, and cornstarch until smooth and well combined.
Return the sliced eggs and cooked chicken to the skillet, combining them with the sautéed vegetables. Pour the prepared sauce over the mixture, stirring to evenly coat all components. Allow the sauce to thicken slightly as the ingredients heat through.
Remove the skillet from the heat and serve the Moo Shu Chicken immediately, pairing it with your choice of sides for a complete meal.