Place dried mushrooms in a small bowl and cover with 1/2 cup very hot water. Let sit for 20 minutes, until mushrooms are full and soft. Drain and discard the liquid. Roughly chop the mushrooms. Set aside.
Heat sesame oil in a medium skillet over medium heat. Add ginger and cook until very fragrant, about 15 seconds. Stir in the five-spice powder, the reconstituted mushrooms and the shiitake mushrooms and cook for 1 minute, stirring constantly. Add carrots, leeks, and scallions. Stir to combine. Add the sherry and stir well. Add the soy sauce and salt and pepper to taste. Transfer to a bowl to cool slightly, 5 to 10 minutes. Stir in the pork.
Line a baking sheet with a parchment paper. Work on 1 dumpling at a time, keeping the remaining wrappers covered with plastic wrap. Place about 2 teaspoons of filling in the middle of a wrapper. Using your fingers, moisten the edges of the wrapper with water and fold in the middle to form a crescent shape. Squeeze the edges together to seal the dumpling. Place filled potsticker on prepared baking sheet and cover with plastic wrap. Repeat with remaining ingredients until all the wrappers are filled.
Heat 1 1/2 tablespoons oil in a cast-iron skillet over medium-high heat for 3 to 4 minutes. Arrange half of potstickers, pleat-side up, snugly together in heated pan. Cook until dumplings are a deep golden brown on the bottom, 1 to 2 minutes. Add 1 cup of hot water, partially cover and cook for 4 to 5 minutes. Reduce heat to medium and cook until bottoms are dark brown, crisp and all the water has evaporated, about 5 to 6 minutes. Slide a spatula under dumplings to loosen. Serve immediately. Wash and dry the pan and repeat cooking process with remaining dumplings and oil.