Make the barbecue sauce: Heat the butter in a heavy saucepan over medium heat. Add the onion and garlic and cook, stirring, until softened, about 10 minutes. Add the ketchup, Worcestershire sauce, mustard, vinegar, brown sugar, chili powder, and hot pepper sauce, and simmer the mixture, stirring occasionally, for 15 minutes. Transfer the sauce to a bowl and let it cool. The barbecue sauce may be made 4 days in advance and kept covered and chilled.
Make the burgers: In a bowl combine well the pork, the bread crumbs, and 1/3 cup of the barbecue sauce and form the mixture into 4 (1-inch-thick) patties. Lightly brush the burgers with oil and season with salt and pepper. Heat a well-seasoned grill pan over medium-high heat. Grill the burgers, turning once, until just cooked through, about 6 minutes per side. Baste the burgers with some of the barbecue sauce and continue to grill, turning and basting occasionally, for 2 minutes more
While the burgers are grilling, make the coleslaw: In a bowl combine the cabbage, carrot, onion, lemon juice, mayonnaise, and salt and pepper, to taste.
Transfer the burgers to the buns and top them with the coleslaw. Serve the remaining barbecue sauce separately.
Recipe courtesy of Gourmet Magazine