1. Combine 2 tablespoons soy sauce and vinegar in a medium bowl. Add carrot and cucumber; toss. Cover and chill 1 hour. Toss. Drain; set vegetables aside. Combine mayonnaise, wasabi, and juice in a small bowl, stirring well.
2. Preheat broiler to high.
3. Cut each bread slice into a 3-inch circle using a round cutter; reserve scraps for another use (such as breadcrumbs or croutons). Lightly coat both sides of bread rounds with cooking spray. Place bread rounds on a baking sheet in a single layer; broil 2 minutes on each side or until toasted. Cool on a wire rack.
4. Combine remaining 1 tablespoon soy sauce, chives, chile paste, and tuna in a large bowl; gently mix until blended. Spread about 1/2 teaspoon mayonnaise mixture on one side of each bread round; top each with 1 lettuce leaf. Divide tuna mixture evenly among sliders, mounding about 2 tablespoons on each. Cut each avocado half into 8 slices. Place 2 avocado slices, 2 cucumber slices, and 2 carrot slices on each slider. Top each slider with remaining bread round, mayonnaise-coated side down. Serve with lime wedges, if desired.