Preheat oven to 162 degrees C. Using a paper towel to help grip, remove and discard skin from chicken thighs. Pat chicken dry and season with salt and pepper. In a very large skillet, heat 2 tablespoons of the oil over medium-high heat. Cook chicken until browned on both sides, 6 to 8 minutes, turning once halfway through. Remove from skillet. Drain and discard fat. Let pan cool slightly.
In the same skillet, heat the remaining tablespoon of oil over medium heat. Add ginger and garlic. Cook and stir just until fragrant, about 30 seconds. Add curry paste; cook and stir 30 seconds more. Add coconut milk, brown sugar, fish sauce, peanut butter, and the water. Whisk to combine. Return chicken to pan; bring to a simmer.
Cover pan (use foil if skillet has no lid) and transfer to oven. Bake for 25 minutes. Remove pan from oven. Carefully turn chicken and stir in sweet pepper and onion. Bake, covered, until chicken is done (80 degrees C), about 10 minutes more. Stir in basil and lime juice and serve over rice.
