Heat 1 tsp sesame oil in large pan over medium-high heat. Crumble the tofu into the pan and add noodles, mushrooms, carrot, cabbage, ginger and garlic and heat for a few minutes. When the tofu and mushrooms are cooked through, add scallions and soy sauce and mix well.
Add 1 Tbsp of the tofu mixture into each wrap. Moisten the edges of the wrap with water, fold in half and press to seal, making a few pinches or folds to create a fan-like shape. Repeat for remaining dumplings.
Add sesame oil and dumplings to pan, a few at a time, and brown on each side (about a minute or two). Add about 1 Tbsp of water to the pan and cover, allowing wraps to steam. Once water evaporates, remove dumplings to a plate. Repeat for remaining dumplings.