Pat the chicken dry with paper towels.
Heat the cooking oil in a pan wide enough to accommodate the chicken legs in a single layer.
Over high heat, brown the chicken on both sides.
Add the ginger, garlic, kaffir lime leaves, chilies (reserve some for garnish) and a tablespoon of fish sauce. Sprinkle in some black pepper.
Pour in the coconut milk. Swirl the pan to distribute the ingredients evenly. Bring to a simmer (do not boil), lower the heat, cover and cook over low heat for about 20 minutes or until the chicken is done and the coconut milk has reduced and thickened.
Turn off the heat. Taste the sauce. Add more fish sauce, if needed. Drizzle in the lime juice. Stir a few times.
Scoop out the chicken legs and transfer to a serving bowl. Pour the sauce and spices around them. Top with sliced scallions and finely-sliced chili. Serve the coconut lime chili chicken hot.