Combine rice vinegar, brown sugar, lime juice, red curry paste, and crushed red pepper in a large zip-top plastic bag. Add chicken; seal and marinate in refrigerator 15 minutes, turning once.
Remove chicken from the bag, reserving marinade. Heat 1 tablespoon oil in a large nonstick skillet or wok over medium-high heat. Add chicken; stir-fry 4 minutes. Remove chicken from pan; keep warm. Add remaining 1 1/2 teaspoons oil to pan. Add onion and carrot; stir-fry 2 minutes. Add mushrooms; stir-fry 3 minutes. Add reserved marinade, scraping pan to loosen browned bits. Add coconut milk and fish sauce; bring to a boil. Reduce heat, and simmer 1 minute. Stir in chicken and salt; cook 1 minute. Top with sprouts and cilantro.