Heat sesame oil on a large wok or skillet over medium. Saute garlic, ginger, and shallot for 2-3 minutes until fragrant. Stir in mushrooms and cook, turning occasionally, for 8-10 minutes until soft and juicy. Combine cabbage and baby shrimp with the rest of the filling. Toss everything with soy sauce.
Place a wonton wrapper on a clean, dry surface. Add about 1 tbsp filling to the center. Dampen the edges (circles don't have edges, I know, but I can't think of another term), then fold over the filling like you would if typing a paper bag. Pinch firmly to seal. Repeat until all filling is used. You may get up to 30 dumplings depending on the size of the wrappers.
In a clean skillet, heat a couple tablespoons of avocado or olive oil over medium-high. Place dumplings flat-side down and brown uncovered for 3-4 minutes. Carefully add 1/4 cup of water, cover, and steam for 5 minutes. Remove to a serving plate. You may need to do multiple batches of dumplings to prevent overlap and get a nice, crispy bottom.
For the sauce, simple whisk together all listed ingredients until combined. Serve dumplings with dipping sauce on the side.