Heat 1 tablespoon vegetable oil in skillet over medium heat; place chicken legs in the pan and cook until golden brown on each side, about 4 minutes per side. Remove chicken from skillet and set aside.
Place coconut milk, water, garlic, jalapeno peppers, cilantro, and green onion into a blender. Cover and puree until smooth. Set aside.
Heat 1 tablespoon vegetable oil in a Dutch oven over medium-high heat. Stir in sliced onion and ginger; cook until sizzling, 1 to 2 minutes.
Sprinkle red curry powder over onions and ginger; cook, stirring constantly, until fragrant and lightly toasted, about 1 minute.
Place chicken legs in a single layer on top of onions and ginger. Pour pureed coconut mixture, lime juice, fish sauce, and brown sugar over the chicken.
Bring mixture to a boil and reduce heat to low. Cover and simmer, skimming off any fat, until chicken legs are tender, about 45 minutes. Remove chicken from the Dutch oven.
Stir potatoes into coconut milk mixture, increase heat to medium-high and bring to a boil. Stir in eggplant; cook until potatoes and eggplant are tender, about 20 minutes.
Stir basil into coconut milk mixture; season with salt and pepper to taste.
Return chicken to coconut milk mixture; reduce heat to medium-low and simmer until chicken is heated through, about 5 minutes.