In a medium bowl, combine the cabbage, carrots, mushrooms, green onions, ginger root, garlic, sesame oil and 2 tablespoons of the vegetable oil.
Mix together until all ingredients are well blended.
Heat a large skillet over medium heat. Add the remaining 1 tablespoon of vegetable oil to the skillet.
Add the vegetable mixture to the skillet and cook for about 5 minutes or until vegetables are tender.
Add 1 tablespoon of soy sauce and mix together.
Spoon about 1 tablespoon of the mixture onto each wonton wrapper. Moisten edges with water and fold in half to form a triangle shape. Press edges together to seal.
Heat a large skillet over medium heat. Add enough vegetable oil to lightly coat the bottom of the pan.
Place potstickers in pan and cook for 2-3 minutes or until golden brown on bottom side. Carefully flip potstickers over and cook for an additional 2-3 minutes or until golden brown.
Add remaining 2 tablespoons of soy sauce to pan and cook for an additional minute or two until liquid is almost evaporated. Serve hot.