Stir together 3 tablespoons soy sauce, 3 tablespoons water, vinegar, 1 teaspoon sesame oil, and sliced chiles in a small bowl.
Heat broiler. Mix together 1 1/2 teaspoons soy sauce, the ginger, garlic, and the chopped scallion and chile in a medium bowl. Mix in turkey until just combined.
Form mixture into 24 balls, dipping hands in water if needed to prevent sticking. Brush meatballs with 1 teaspoon sesame oil on a baking sheet. Broil 8 inches from heat source until golden brown and cooked through, 5 to 7 minutes.
Meanwhile, heat remaining teaspoon sesame oil in a large nonstick skillet over medium-high heat. Cook mushrooms until just starting to brown, about 2 minutes. Reduce heat to medium. Cook, covered, until tender, about 3 minutes more. Add remaining 1 1/2 teaspoons soy sauce. Toss to coat.
Arrange lettuce leaves on a platter, and top each with 1 meatball. Top with mushrooms, and sprinkle with sliced scallion. Serve with dipping sauce.
Dipping sauce can be refrigerated for up to 1 day. Shaped meatballs can be refrigerated for up to 1 day.