In a bowl, mix turkey, egg, mushrooms, cabbage, water chestnuts, green onions, cornstarch, ginger, 3 tablespoons soy sauce, oyster sauce, wine, salt, sugar, and pepper until well blended.
To assemble each dumpling, fill a small bowl with water. Place one wrapper on a flat surface; cover remaining wrappers with plastic wrap to keep pliable. Place a scant tablespoon of filling in center of wrapper. With your fingers, moisten edge of wrapper all the way around with water, then bring opposite sides together over filling and pinch edges together only in the center. On front side of dumpling, make two pleats on each side of center, folding to the center, and press all edges to seal. Set dumpling, seam side up so dumpling sits flat, on a lightly floured baking sheet. Cover with plastic wrap while you fill remaining wrappers (see notes).
To cook each batch, set a 10- to 12-inch nonstick frying pan over medium-high heat (see notes). When hot, coat pan bottom with 1 tablespoon vegetable oil. Set dumplings, seam up and slightly apart, in a single layer in pan. Cook until bottoms are golden brown, 3 to 5 minutes. Add 1/3 cup water, cover pan tightly, reduce heat to medium-low, and cook until filling is no longer pink in the center (cut to test), 3 to 6 minutes (10 to 11 minutes if frozen). Uncover and, if there is still liquid in pan, continue cooking over medium-high heat until the liquid has evaporated.
Using a wide spatula, transfer dumplings to a heatproof platter. Serve, or cover and keep warm in a 200° oven. Repeat to cook remaining dumplings.
Serve with soy sauce, rice vinegar, and chili oil for dipping.