Heat sesame 1 T oil in skillet on high. Add veggies and ginger. Cook, stirring occasionally, about 5 minutes until veggies have softened. Add tamari, green onions, sesame seeds and cook another 2-3 minutes until liquid has evaporated. Cool.
Lay out wrappers and brush edges with water. Place 1 T filling on each wrapper and fold over and pinch to seal.
Heat skillet with remaining oil to medium high. Add pot stickers and cook until bottoms are light brown, 2-3 minutes. Turn over. Pour a scant 1/4 cup water into pan and cover tightly. Turn heat down to medium and steam dumplings for 3 minutes. Serve immediately with sauce