Heat 1 tablespoon of cooking oil in a large skillet or wok over medium-high heat. Add the minced garlic and grated ginger, and sauté for about 30 seconds until fragrant.
Add the sliced shiitake mushrooms and cook for 3-4 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 shredded chicken to the skillet, followed by the soy sauce, hoisin sauce, sesame oil, salt, and pepper. Stir well and cook for an additional 2-3 minutes until everything is heated through.
Remove the skillet from heat and stir in the chopped green onions.
Warm the tortillas or Chinese pancakes in a separate skillet or microwave.
To serve, spoon the Moo Shu chicken mixture onto each tortilla or pancake, roll it up, and enjoy!