Barbecue Sauce

1. In a saucepan over medium heat, melt the butter. Add the chopped onion and cook until translucent. Add the garlic and sauté for about 2 minutes longer.
2. Add the ketchup, lemon juice, Worcestershire sauce, molasses, mustard, and cayenne pepper. Cover the pan with a lid slightly ajar to reduce splatters. Simmer, stirring occasionally, for about 10 minutes. Set aside.

Coleslaw

1. Put the shredded cabbage and carrots in a large bowl.
2. whisk the mayonnaise with vinegar, sugar, celery seeds, onion powder, and 1/4 teaspoon of freshly ground black pepper in a small bowl. Add the dressing to the cabbage mixture and toss to blend ingredients. Add salt to taste or as desired. Cover and refrigerate until serving time.

Burgers

1. Prepare the gas or charcoal grill for direct heat, with burners on high or preheat an even layer of coals.
2. Alternatively, oil a grill pan and heat over high heat on the stovetop.
3. Separate the ground beef into four 6-ounce portions and shape into burgers. With your thumb, make a slight indention in the center of each. Rub the patties with olive oil and sprinkle with salt and freshly ground black pepper.
4. Grill the burgers over high heat for 4 to 5 minutes or until they easily release from the grates. Flip and cook for another 4 to 5 minutes on the other side. Brush with sauce and cook for 1 to 2 minutes until they reach at least 155 F in the center. According to the USDA, the minimum safe temperature for ground meat is 160 F. The burgers should increase to that temperature off the heat. If you like a medium burger, cook them to 145 to 150.
5. Meanwhile, toast or grill the split hamburger buns.
6. Assemble the burgers in the buns with more of the barbecue sauce and a lettuce leaf, if desired. Top the burgers with a generous spoonful of coleslaw or serve the coleslaw along with the other toppings at the table.