Step 1Soak the mushrooms in boiling water for 20 minutes. Drain well, then remove any stems and cut the caps into a one-eighth-inch dice. Place in a mixing bowl.
Step 2Carefully wash the spinach leaves in several changes of water. With just the water clinging to the leaves, add to a large pot. Cover and cook over high heat just until the leaves wilt, about 2 minutes. Drain well, squeeze dry and finely chop. Add to the mushrooms.
Step 3Beat 1 egg in a small bowl with salt and pepper to taste. Heat 1 teaspoon sesame oil in a small skillet over medium heat. Add the beaten egg and swirl to coat the bottom of the pan. Cook until set, flipping once. Turn out onto a plate and repeat with the second egg. Cut these flat omelets into a one-fourth-inch dice and add to the bowl with spinach.
Step 4Add the tofu, green onions, garlic, ginger and soy sauce. Taste and adjust the seasonings if necessary.
Step 5To stuff the dumplings, mound about 2 teaspoons of the filling in the center of each wrapper and brush the outer edges lightly with water to moisten. Fold the wrapper in half, into a half-moon shape, then pinch it together at the center of the top edge. Pleat the edges of the wrapper to enclose the filling. Repeat with the remaining dumplings.
Step 6To pan-fry the dumplings, heat a 10-inch skillet with a lid over medium heat. Add 2 tablespoons oil and swirl the pan. When the oil is hot, place 8 to 10 dumplings in the pan, leaving room between each, and cook until the bottoms just start to brown, about 2 minutes. Add one-third cup water and reduce the heat to medium-low. Cover the pan and cook until the dumplings are soft on top, crisp on the bottom and the filling is firm, 5 to 7 minutes. Repeat with the remaining dumplings.
Step 7To steam the dumplings, use a steamer of your choice and bring 1 to 2 inches of water to a simmer (make sure the water level does not reach the bottom of the steamer). Lightly oil the steamer rack and place the dumplings on the rack about 1 inch apart from one another (the dumplings will need to be steamed in batches). Place the steamer over the simmering water and cover. Steam the dumplings until the filling is firm and the wrapper is slightly translucent, about 10 to 15 minutes. Remove the steamer from the water and gently remove the dumplings to a warm place until all of the dumplings are cooked. Serve warm with a dipping sauce.