In a 10-12-inch frying pan over medium-high heat, cook bacon, in batches if necessary, turning occasionally, until browned on both sides and crisp, 8 to 10 minutes. With tongs, transfer to paper towels to drain. When cool, break each slice in half.

Meanwhile, in a large bowl, beat eggs, milk, onion, garlic, 1 1/4 teaspoons salt, and 1/2 teaspoon pepper to blend. Add beef and bread crumbs and mix gently. Form into 10 equal patties, each about 4 1/2 inches wide.

Brush onion slices lightly on both sides with bacon fat in pan. Discard remaining fat.

Lay patties and onion slices on a barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook, turning meat and onions once to brown on both sides, until a thermometer inserted in center of thickest part of meat registers 160° (no longer pink in center), 6 to 8 minutes total.

About 2 minutes before patties are done, top each with a slice of cheese. Also lay bun halves, cut side down, on grill and toast 1 to 2 minutes.

Set bun bottoms on a platter or plates. With a wide spatula, transfer patties to bun bottoms. Top each with a grilled onion slice, 2 pieces of bacon, and a spoonful of special slaw. Add salt and pepper to taste. Cover with bun tops.