1. In large saucepan, heat the butter over medium-high heat until melted and bubbly. Add the onion, sweet green pepper and garlic; saute for 5 minutes or until the vegetables are softened.
2. To the saucepan, add the tomatoes, ketchup, water, red-wine vinegar, dark-brown sugar, mustard, paprika, cumin, hot-pepper sauce, salt and black pepper; stir to mix well. Bring to a simmer over medium-high heat. Cover; lower the heat to medium-low. Simmer, adjusting the heat as needed to maintain a simmer, stirring occasionally, for 30 minutes.
3. Uncover saucepan; cook, stirring occasionally, over medium-low heat for 30 minutes. Remove from heat until ready to use. Sauce may be prepared ahead and refrigerated, tightly covered, for up to 3 days.
4. Prepare outdoor grill with medium-hot coals, or heat gas grill to medium-high. Lightly brush grill grid with vegetable oil. (Or cook chicken using oven method, below.)Chicken:
5. Reserve half the BBQ sauce (about 2-1/4 cups) for serving on the side and use the other half for cooking chicken.
6. Grill the chicken, uncovered, for 5 minutes on each side. Spoon the BBQ sauce generously over the chicken. Cover the grill; grill the chicken for 10 minutes. Spoon on more sauce; turn the chicken over and continue to grill, covered, turning the chicken pieces over every 5 to 10 minutes, spooning on more sauce each time, until the internal temperature registers 170 degrees F on an instant-read meat thermometer inserted in the thickest parts without touching any bone. The total cooking time will be about 60 to 70 minutes; do not spoon any sauce over the chicken quarters during the last 10 minutes of grilling.
7. Serve chicken with reserved BBQ sauce.