This dish features a Vietnamese-style sauce known as "nuoc cham" [noo-ahk CHAHM] that traditionally includes fish sauce, lime juice, chiles, and sugar. Your avocados should be ripe but still slightly firm so they'll slice easily and not break apart as they grill. Leaving the skin on helps them maintain their shape—remove skin before serving, if you like.
Combine first 5 ingredients in a small bowl. Reserve 3/4 cup juice mixture. Stir carrot and chile into remaining 1/2 cup juice mixture, and set aside.
Peel shrimp, leaving tails intact. Starting at tail end, butterfly each shrimp, cutting to, but not through, backside of shrimp. Combine reserved 3/4 cup juice mixture and shrimp in a large bowl, tossing to coat; cover and marinate in refrigerator 1 hour, tossing occasionally. Remove shrimp from bowl, reserving marinade.
Prepare grill for medium-high heat.
Place reserved marinade in a small saucepan; bring to a boil. Reduce heat, and simmer 5 minutes. Cool slightly.
Thread 3 shrimp onto each of 12 (10-inch) skewers. Place shrimp on a grill rack coated with cooking spray. Grill shrimp 2 1/2 minutes on each side or until done, basting frequently with cooked marinade.
Cut 3 avocado halves in half lengthwise; peel and dice remaining avocado half. Brush cooked marinade over mango and avocado wedges; coat with cooking spray. Arrange in a single layer on grill rack coated with cooking spray. Coat lime wedges with cooking spray; place on grill rack. Grill fruit 2 minutes on each side or until marked but not soft, basting frequently with marinade.
Place 1 lettuce leaf on each of 6 salad plates; top each leaf with 2 mango slices, 2 lime wedges, and 1 avocado wedge. Place 2 skewers on each plate; sprinkle evenly with diced avocado. Sprinkle with cilantro, if desired. Serve with carrot mixture.