In a medium bowl, combine ground chicken, shredded carrots, shredded green cabbage, garlic, chopped green onions, hoisin sauce, sesame oil, and egg. Mix until well incorporated.
Lay a wonton wrapper on a solid surface. Add a heaping teaspoon of the chicken mixture to the center. Wet the outside edges of the wrapper with water and fold it over to seal tightly, forming a half-moon shape.
Heat vegetable oil in a large skillet over medium heat. Place the potstickers in the skillet and cook on each side for 2-3 minutes until golden brown.
Add 2 tablespoons of water to the skillet and cover. Cook for an additional 3-4 minutes until the water evaporates and the potstickers are cooked through.
Serve immediately with soy sauce or your favorite dipping sauce.
