Combine 2 tablespoons oil, soy sauce, sugar, and vinegar in a bowl, stirring with a whisk.
Heat remaining 1 tablespoon oil in a large nonstick skillet over medium. Add turkey; cook 7 minutes or until lightly browned, stirring to crumble. Add 1/4 cup soy sauce mixture; cook 4 minutes or until liquid is absorbed.
Place about 3 tablespoons turkey mixture in each lettuce leaf; top evenly with cucumber, carrots, and peanuts. Drizzle evenly with remaining soy sauce mixture. Sprinkle with mint.