In a large skillet or wok, heat the vegetable oil over medium-high heat. Add the garlic and sauté for about 30 seconds until fragrant.
Add the shiitake mushrooms and cook for 2-3 minutes until they start to soften.
Stir in the napa cabbage and carrots, cooking for an additional 3-4 minutes until the vegetables are tender.
Add the shredded chicken, soy sauce, hoisin sauce, and sesame oil to the skillet. Stir well to combine and heat through, about 2-3 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.