Combine ingredients for dipping sauce and set aside.
Prepare the dumplings by heating a large, non-stick wok over medium-high heat. Swirl in the canola and sesame oils and add ginger and garlic. Stir fry for about 10 seconds and stir in mushrooms and cabbage for 3 minutes.
Add the sake and cook, stirring, for 3 - 4 minutes until liquid evaporates. Next add the scallions and the carrot and stir-fry about 2 minutes. Remove from heat and add soy sauce. Allow mixture to cool.
Place 1 teaspoon of the vegetable mixture in center of a wrapper. Moisten edges of wrapper with water and fold unto half circles. Place on baking sheet sprinkled with cornstarch. Keep wontons covered with a damp towel to prevent them from drying out.
Spray skillet with nonstick spray heat over medium-high heat. Add 10 dumplings to the skillet and cook until browned on one side. Add 1/3 cup water to the skillet and cover with lid. Reduce heat to medium and cook until liquid evaporates (3 -4 minutes).