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1
In large bowl, toss together cabbage and 3/4 teaspoon salt and set aside for 30 minutes. Transfer to clean dish towel or cheesecloth, gather ends together, and twist to squeeze out as much water as possible. Wipe bowl clean, then return cabbage to it. Add pork, ginger, carrots, scallions, and garlic and stir to combine.
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2
In small bowl, whisk together soy sauce, sesame oil, and egg, then stir into cabbage-pork mixture. Stir in pepper and remaining 1/8 teaspoon salt.
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3
On dry surface, lay out 1 gyoza wrapper, keeping remaining wrappers covered with dampened cloth or paper towel. Spoon 1 1/2 teaspoons filling into center, then moisten halfway around edge with wet finger. Fold moisture-free half of wrapper over moistened half to form open half-moon shape. To seal, using thumb and forefinger of one hand, form 6 tiny pleats along unmoistened edge of wrapper, pressing pleats against moistened border to enclose filling. Moistened border will stay smooth and will automatically curve in semicircle. Stand dumpling, seam-side up, on baking sheet and gently press to flatten bottom. Cover loosely with dampened cloth or paper towel. Form remaining dumplings in same manner.
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4
In 10-inch, lidded, non-stick skillet over moderately high heat, heat oil until hot but not smoking, then remove from heat and arrange pot stickers in tight circular pattern standing up in oil (they should touch one another). Cook, uncovered, until bottoms are pale golden, 2 to 3 minutes. Add 1/2 cup water, tilting skillet to distribute, then cover tightly with lid and cook until liquid has evaporated and bottoms of dumplings are crisp and golden, 7 to 10 minutes. Add 2 tablespoons more water if skillet looks dry before bottoms are browned. Remove lid and cook, shaking skillet to loosen pot stickers, until steam dissipates, 1 to 2 minutes. Invert large plate with rim over skillet. Using pot holders, hold plate and skillet together and invert skillet. Remove skillet and serve pot stickers warm.
