Mushroom Potstickers

Name: Mushroom Potstickers
Ingredients: 1.5 ounces dried mushrooms, such as maitake, wood ear, or oyster (see Notes); 2 to 4 tbsp. vegetable or canola oil, divided; 8 ounces fresh shiitake mushrooms, finely chopped; 1 tablespoon minced shallot; 1 garlic clove, minced; 1/2 teaspoon salt; About 1 tbsp. soy sauce; About 2 tsp. toasted sesame oil; 1 egg, beaten; 36 gyoza (round Japanese potsticker) wrappers (see Notes)