Heat a large skillet or wok over medium-high heat and add a tablespoon of cooking oil.
Add the minced garlic and ginger, sautéing for about 30 seconds until fragrant.
Add the sliced shiitake mushrooms and cook for 2-3 minutes until they start to soften.
Stir in the shredded napa cabbage and julienned carrots, cooking for another 3-4 minutes until the vegetables are tender.
Add the cooked chicken to the skillet, along with the soy sauce, hoisin sauce, and sesame oil. Stir well to combine and heat through, about 2 minutes. Season with salt and pepper to taste.
Warm the flour tortillas or moo shu pancakes in a separate pan or microwave.
Serve the chicken mixture in the tortillas or pancakes, garnished with chopped green onions.