Put dried mushrooms in a bowl, cover with 2 cups boiling water, and let sit until tender, about 30 minutes. Drain, reserving soaking liquid; finely chop mushrooms.
Heat 1 tbsp. oil in a large frying pan over medium-high heat. Add shiitakes and cook, stirring occasionally, until they start to release their liquid. Add shallot, garlic, and salt. Cook, stirring, until fragrant, about 30 seconds. Add dried mushrooms and 1/2 cup soaking liquid. Cook, stirring, until liquid has evaporated, 3 minutes.
Transfer mixture to a large bowl and let cool to lukewarm. Stir in soy sauce and sesame oil; add more if you like. Stir in egg.
On a clean work surface, lay down a wrapper. Put about 1 tbsp. mushroom mixture in the center. Dampen edges of wrapper with water, fold in half, and seal. Put dumpling on a baking sheet or dry work surface. Continue with rest of wrappers and filling.
In a large nonstick frying pan, heat 2 tsp. oil over medium-high heat. Place as many dumplings in pan as will fit without touching. Cook until bottoms of dumplings are brown, 2 to 3 minutes. Add 3/4 cup reserved soaking liquid or water, cover, and cook until liquid is absorbed and bottoms are sizzling again, 3 to 5 minutes. Transfer to serving dish and repeat with remaining dumplings. Serve hot or warm.
Note: Nutritional analysis is per potsticker.