To prepare dipping sauce, combine first 5 ingredients, stirring well with a whisk.
To prepare dumplings, place mushrooms, onions, carrot, ginger, and vinegar in a food processor; pulse until coarsely chopped, scraping sides. Combine mushroom mixture and egg whites in a large bowl; add turkey, stirring until combined.
Working with 1 gyoza skin at a time (cover remaining skins to keep from drying), spoon about 1 tablespoon turkey mixture into center of each skin. Moisten edges of skin with water. Fold in half, pinching edges together to seal. Place dumpling on a baking sheet (cover loosely with a towel to keep from drying). Repeat procedure with remaining skins and turkey mixture.
Heat 1 teaspoon vegetable oil in a large nonstick skillet over medium-high heat. Arrange half of dumplings in pan; cover and cook 5 minutes or until lightly browned. Turn dumplings; add 1/4 cup water. Cover and simmer 5 minutes. Remove from pan; keep warm.
Wipe pan dry with a paper towel. Repeat procedure with remaining vegetable oil, dumplings, and water. Serve warm with dipping sauce.