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In a small bowl, cover the dried mushroom with hot water and let stand until softened, about 5 minutes. Drain and coarsely chop the mushroom. Peel and devein the shrimp, reserving 20 with their tails. Coarsely chop the remaining shrimp.
In a food processor, finely chop the carrot. Add the onion and mushroom and finely chop. Add the chopped shrimp, fish sauce, oil, salt and pepper; process just until blended.
If using spring roll wrappers, cut them in half diagonally to form 2 triangles. Lay 2 of the triangles on a work surface with the long side facing down; keep the remaining wrappers covered with a damp towel. Place a whole shrimp in the center of each wrapper with the tail sticking up beyond the tip of the triangle. Spread 1 scant tablespoon of the chopped filling over each shrimp and brush the edges of the wrappers with a little beaten egg. Fold the longest side of each wrapper up and over the shrimp, wrapping the ends tightly around it. If using wonton wrappers, lay one on a work surface and brush the edges with egg. Set a shrimp in the center, with the tail sticking out of the wrapper. Spread with 1 tablespoon of the filling and cover with another wrapper. Repeat with the remaining wrappers, shrimp and filling.
In a heavy saucepan, heat the vegetable oil to 350° over moderately high heat. Fry 3 or 4 dumplings at a time, turning once, until golden brown and crisp, 4 to 5 minutes. If necessary, lower the heat so that the oil won't smoke. Drain the dumplings briefly on paper towels, then transfer to a platter and keep warm. Serve with a bowl of Nuoc Cham.
The spring roll dumplings can be refrigerated for 1 day. Don't make the wonton dumplings ahead.