Heat 2 teaspoons vegetable oil in a large nonstick skillet over medium-high heat. Add cabbage; cook 9 minutes or until lightly browned, stirring frequently. While cabbage cooks, add 1/2 cup water, 1 tablespoon at a time, to keep cabbage from sticking to pan. Spoon cabbage into a medium bowl; let cool completely. Add chicken and next 6 ingredients (chicken through garlic) to bowl; stir well.
Working with 1 won ton wrapper at a time (cover remaining won ton wrappers to keep them from drying out), spoon about 1 tablespoon chicken mixture into the center of each wrapper.
Moisten edges of wrapper with water, and bring 2 opposite corners to center, pinching points to seal. Bring the remaining 2 corners to center, pinching points to seal. Pinch 4 edges together to seal. Place pot stickers on a large baking sheet sprinkled with cornstarch; cover loosely with a towel to keep them from drying out.
Heat 2 teaspoons vegetable oil in a large nonstick skillet over medium heat. Place half of pot stickers in bottom of skillet; cook 3 minutes or until bottoms are lightly browned. Add 1/2 cup water to skillet; cover and cook 3 minutes or until liquid is absorbed. Place pot stickers on a serving platter; set aside, and keep warm.
Wipe skillet with a paper towel. Repeat procedure with remaining vegetable oil, remaining pot stickers, and remaining water.