In a medium bowl, combine the lamb, mushrooms, ginger, garlic, soy sauce, green onions, and a pinch of salt and black pepper.
Add about 1 teaspoon lamb filling to each wonton wrapper. Brush water around the edges of the wrapper then fold the edges up around the filling and pinch to seal. Add sesame seeds to a small bowl or shallow dish. Brush the bottoms of the dumplings with water then dip in sesame seeds.
Heat 1 tablespoon sesame oil in a medium pot over medium-high heat. Once hot, add the shallots and cook until softened, 3-4 minutes. Add ginger and garlic and cook another minute, until fragrant. Add the broth, soy sauce, and the rice vinegar. Season to taste with pepper. Bring to a simmer over medium heat. Keep warm.
Heat 2 tablespoons vegetable oil in a large skillet set over medium-high heat. Once hot, add the dumplings and cook until the bottoms are golden brown, 4-5 minutes. Carefully add 1/4 cup water to the skillet, and cover immediately. Reduce heat to medium-low and let the dumplings steam for 5-6 minutes*.
Ladle broth into bowls and top with wontons, green onions and chili oil. You can also serve the dumplings on their own with gyoza dipping sauce or potsticker sauce.
* Note: USDA recommends ground lamb reach an internal temperature of 160F.