Whisk dark brown sugar, rice vinegar, water, fish sauce, soy sauce, garlic, ginger, and crushed red pepper together in a bowl until sugar is completely dissolved.
Heat oil in a skillet over high heat. Add chicken; cook until skin is golden brown, about 5 minutes. Pour in 1/2 cup of the brown sugar mixture. Cook and stir until sauce is thickened, about 5 minutes more.
Pour in remaining sauce; cook until chicken is no longer pink at the bone and the juices run clear, 10 to 12 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C). Top with green onions.