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In a small skillet, heat 1 teaspoon vegetable oil over medium-high. Add 2 tablespoons ginger and scallions and cook, stirring, until scallions are slightly softened and mixture is fragrant, 2 minutes. Remove from heat. In a food processor, pulse shrimp until finely chopped. Transfer to a bowl and add ginger mixture, curry paste, and fish sauce; stir to combine.
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Working with 1 wonton wrapper at a time, place 1 heaping teaspoon shrimp mixture in center. With a wet finger, moisten edges of wrapper, then bring all 4 corners together, pinching at top to seal. (Cover filled dumplings with a kitchen towel as you work.)
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In a large nonstick skillet, heat 1 teaspoon vegetable oil over medium-high. Add dumplings, seam side up, and cook until golden brown on bottom, 1 minute. Carefully add 1/2 cup water to skillet, cover, and cook until water is almost evaporated and dumplings are tender, 3 minutes. Uncover and cook until water evaporates, 1 minute (reduce heat to medium if dumplings begin to overbrown). Meanwhile, in a small bowl, combine 1 tablespoon ginger, soy sauce, and sesame oil. Serve dumplings with dipping sauce.
