Put a baking sheet on the middle oven rack; preheat to 500˚. Season the chicken all over with salt and pepper in a large bowl. Add the curry powder, paprika and half the melted butter; toss.
Toss the turnip and carrots with the remaining melted butter in another bowl; season with salt and pepper. Spread out the vegetables and chicken, skin-side up, on the hot baking sheet. Roast until the chicken is cooked through and the vegetables are tender, about 25 minutes.
Meanwhile, puree the buttermilk, cilantro, mint, serrano, 1 teaspoon chutney and the lemon juice in a blender until smooth. Season with salt and pepper.
Preheat the broiler. Stir the remaining 1/4 cup chutney with 1 teaspoon water in a small bowl. Brush on the chicken and broil until lightly browned, 2 to 3 minutes. Divide the chicken and vegetables among plates. Serve with the herb sauce.