1. Remove and discard chicken skin. Rinse chicken and pat dry.
2. Place a 10- to 12-inch nonstick frying pan over high heat. Add oil and swirl to coat pan; when hot, add chicken and brown on each side, about 5 minutes total. Transfer chicken to plate.
3. Meanwhile, trim and discard root end and coarse outer lemon grass leaves. Cut the stalk into 4-inch sections and crush with a mallet or the blunt edge of a knife.
4. To pan, add coconut milk, curry paste, and sugar. Stir until mixture is smoothly blended. Stir in 1 tablespoon fish sauce and add lemon grass, ginger, and garlic. Heat until boiling.
5. Set aside 4 of the nicest basil leaves; add remainder to pan along with chicken. Cover and simmer over low heat for 15 minutes. Turn chicken over, cover, and continue to simmer until chicken is no longer pink at bone (cut to test), 10 to 15 minutes longer.
6. Lift chicken from sauce and put on a platter; keep warm. With a slotted spoon, remove lemon grass, garlic, ginger, and basil, if desired. If you want the sauce thick enough to coat chicken in a velvety layer, boil and stir to reduce slightly. Pour sauce over chicken and sprinkle with remaining basil leaves. Season to taste with fish sauce.