In a medium bowl, whisk together mayonnaise, ketchup, mustard, relish, and 1 teaspoon black pepper. Stir in 2 tablespoons rendered bacon fat. Set aside.
To make the burgers:
Form ground beef into 4 patties, about 6 ounces each, slightly wider than the buns. Press the center of each patty to make a slight indentation with your fingertips. Season liberally with salt and pepper on all sides. Set aside.
Brush hamburger rolls on inside surfaces with rendered bacon fat and set aside.
Heat 12-inch skillet over high heat. Add remaining rendered bacon fat to skillet. Add burger patties and cook, turning occasionally, until crusted. The center of each burger should register at least 120 degrees on an instant-read thermometer.
Top with cheese and continue cooking until cheese is melted and burgers register 135 degrees for medium or 145 degrees for medium well. Transfer to a large plate.
Wipe out skillet and heat over medium-high heat. Add burger buns and toast until golden, about 3 to 5 minutes.
Spread burger sauce on both sides of burger buns. Place patties on each bottom bun. Top with bacon, lettuce, tomato, and bun tops.