Sherry Vinegar-Brown Sugar Baste: Place medium saucepan on side burner over medium high heat. Add the oil, shallots and garlic and cook until soft. Add the paprika and cook for 2 minutes. Add the sherry vinegar and cook until reduced by 3/4. Add the brown sugar, tomatoes, and stock. Season with salt and pepper and bring to a boil. Reduce the heat to medium-low and simmer for 30 minutes. Transfer the mixture to a food processor and process until smooth. Return the mixture to a clean saucepan and cook until thickened, about 20 minutes. Let cool to room temperature and reserve 1/2 cup.
Chicken: Preheat grill to medium. Brush chickens with olive oil and season with salt and pepper to taste. Grill the chicken, skin-side down, until golden brown, about 4 to 5 minutes. Turn the chicken over and continue grilling until cooked through, 15 to 20 minutes, turning occasionally. Baste with the barbecue baste every 3 minutes. Place the chicken on a large platter and cut in half. Brush with some of the reserved sauce.