In this warm salad, grilled chicken sits on a bed of rice vermicelli garnished with fresh mint and clantro.
Soak vermicelli, if using, in a bowl of cold water for 15 minutes. Meanwhile, in a small bowl, combine chopped chile pepper or red-pepper flakes, sugar, lime juice, 2 tablespoons of the vinegar, and fish sauce. Stir well to dissolve sugar. Set aside.
Bring a pot of lightly salted water to a boil. Drain vermicelli and add to boiling water; cook for 2 minutes. If using somen, add dry noodles to boiling water and cook for 3 minutes. Drain noodles and transfer to a bowl of cold water. Set aside.
Peel carrots and, using a sharp zester, cut into long, thin threads. Use a vegetable peeler to "peel" entire cucumber into long, thin strips. Cut strips lengthwise into thinner strips, about 1/4 inch wide.
Combine carrots, cucumber, cabbage, lettuce, and bean sprouts in a mixing bowl and toss with remaining vinegar. Set aside.
Remove tenderloin (meat attached to underside of breast) from chicken breasts; save for another use. Place breasts between 2 pieces of plastic wrap. Using the flat side of a meat tenderizer or cleaver, pound breasts 1/4 inch thick.
Heat grill or grill pan and brush with oil. Season chicken on both sides with salt and pepper, and grill until cooked through and juices run clear, about 4 minutes per side. Remove from grill and allow to sit for 1 to 2 minutes.
Drain noodles well. Divide salad mixture among 4 plates. Top each with a quarter of the noodles. Slice chicken breasts lengthwise into 1/2-inch-thick strips and place on salads. Set out fish-saucemixture, mint and cilantro leaves, and chopped peanuts in separate bowls, for dressing and garnishing individual salads.
Fish sauce is available in the international-foods section of many supermarkets.