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Lay the wrappers on a work surface. Place 1/2 tablespoon of the filling in the center of each wrapper. Avoid getting any filling on the edges of the wrapper, which would prevent them from sealing properly. Paint the circumference of the wrappers with the egg wash using a finger or pastry brush. Fold each wrapper in half to form a half-moon shape and seal by pressing between your fingers. Starting at the center, make 3 pleats, working toward the bottom-right corner. Repeat, working toward the bottom-left corner. Press the folded edges of the dumplings gently on the work surface to flatten the bottoms and help them stand.
Heat a large, nonstick lidded saute pan over high heat. Add the oil and swirl to coat the pan. When the oil is hot, add the dumplings flattened-bottoms down, in batches if needed, and cook without disturbing until brown, 3 to 4 minutes. Add about 1/2 cup water and immediately cover the pan to avoid splattering. Lift the cover and make sure about 1/8 inch water remains in the pan; add a little more if not. Steam until puffy yet firm and the water has evaporated, 8 to 10 minutes. If the water evaporates before the dumplings are done, add more in 1/4-cup increments. If they seem done, but water remains in the pan, drain it and return the pan to the stovetop to evaporate any remaining liquid. Continue to cook over high heat to allow the dumplings to re-crisp on the bottom, 2 to 3 minutes. Be careful not to burn them.
Transfer the dumplings to a platter and serve with the dipping sauce in individual small ramekins.