In a large dutch oven over medium heat, add 2 tablespoons (28 grams) of sesame oil and heat until fragrant (approximately 1 minute).
Add the garlic and ginger and cook for 2 minutes, stirring often.
Add the mushrooms and tofu and cook for 5 - 8 minutes, stirring often. The mushrooms and tofu will begin to release water as they cook down.
Add the carrots, cabbage, soy sauce, and red pepper flakes and cook for 5 minutes, stirring often.
Add the green onions and stir while cooking.Continue to cook until the majority of liquid has cooked off. Do not allow the vegetables to burn and stick to the bottom of the pan.
Remove the pot from the heat and transfer the dumpling filling to a large bowl to cool.
Place a single dumpling wrapper on a clean kitchen surface and scoop approximately 1 tablespoon of dumpling filling into the center of the wrapper. Have a small bowl of water nearby to dip your finger into. Using the water on your finger, drag it across half of the outside of the dumpling wrapper around the dumpling filling. Fold the dumpling wrapper in half and lightly press the ends together, pressing out any air from the dumpling before sealing it completely. You can now crimp the dumpling to create pleats along the top of the wrapper or join and pinch the two edges together to create a hugged pillow shape.
Place a large non-stick skillet over medium-high heat and add the remaining 1 tablespoon (14 grams) of sesame oil. Heat the oil until fragrant (approximately 1 minute).
Place the dumplings in a single layer within the skillet. Cook until the bottoms of the dumplings are a toasted golden brown color.
Pour approximately ¼ cup (60 grams) of water evenly over the dumplings in the skillet and cover the pan with a lid to steam. Once the water has cooked off, turn the heat off and remove the dumplings from the pan.
Add all sauce ingredients to a small bowl and stir to combine. Taste and add any additional soy sauce, sugar, or chili oil if preferred.
Serve cooked dumplings with sauce and enjoy. Store cooked dumplings or dumpling filling in a food-safe airtight container or resealable bag in the fridge for up to 4 days or in the freezer for up to 2 months.
