Heat a large skillet or wok over medium-high heat and add 1 tablespoon of cooking oil.
Add the minced garlic and ginger, stirring 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 napa cabbage and carrots, cooking for another 3-4 minutes until the vegetables are tender but still crisp.
Add the shredded chicken, soy sauce, hoisin sauce, and sesame oil to the skillet. Stir well to combine and heat through for about 2 minutes. Season with salt and pepper to taste.
Warm the tortillas or moo shu pancakes in a separate pan or microwave.
Serve the chicken mixture in the warm tortillas or pancakes, garnished with sliced green onions.