In a wok, heat 2 tablespoons of the oil until smoking. Add the lemongrass, onion, garlic and ginger and stir-fry over high heat until the garlic is golden, about 2 minutes. Add the remaining 1 tablespoon of oil. Add the chicken and stir-fry until the chicken turns white, about 3 minutes. Add the carrot, the remaining 2 tablespoons of fish sauce, the cayenne and 3 tablespoons of the caramel sauce and cook over moderate heat, stirring, until the chicken is glazed, about 4 minutes. Stir in the broth, simmer for 30 seconds and serve.