Heat the oil in a medium saucepan over medium heat. Add the garlic, onion, paprika, chili powder and smoked paprika and cook, stirring occasionally, until the onions soften, about 8 minutes. Add the ketchup, vinegar, butter, Worcestershire and 1 cup water. Bring to a simmer and cook, stirring occasionally, until slightly thickened and the flavors have blended, about 10 minutes.
Pat the chicken dry and season all over with salt and pepper. Arrange the chicken skin-side down on a rimmed baking sheet. Pour the barbecue sauce over the chicken and cover with foil. Bake about 25 minutes. Remove the foil and continue to bake until the chicken is beginning to brown in spots, about 15 minutes. Turn the chicken skin-side up and bake until golden and browned in spots, about 15 minutes more. Toss to coat with the sauce on the baking sheet and serve.