Preheat oven to 400 degrees.
Heat olive oil in a large ovenproof skillet over medium-high heat. Season chicken with salt and pepper and place in skillet, skin-side down. Cook until skin golden brown and crisp, 4 to 5 minutes. Turn chicken and transfer skillet to oven. Cook until chicken is cooked through, 35 to 40 minutes. Transfer chicken pieces to a platter and set aside.
Place skillet over medium heat; add shallots and garlic. Cook, stirring, until softened, about 30 seconds. Add wine and stir to combine. Cook until liquid is reduced to about 2 tablespoons, 3 to 4 minutes.
Add chicken stock and cook until thickened, 5 to 7 minutes. Stir in chives, parsley, basil, and butter; season with salt and pepper. Return chicken to skillet. Using a spoon, coat chicken pieces with sauce. Serve immediately.