Cook bacon in a skillet over medium heat until crispy, then drain on paper towels and reserve 1 tablespoon of bacon fat.
Finely chop the bacon and mix it with mayonnaise, Dijon mustard, garlic powder, smoked paprika, lemon juice, and the reserved bacon fat. Season to taste and chill for 15 minutes.
Divide ground beef into 4 loose balls (about 5–6 ounces each) and keep chilled. Brush buns with bacon fat or butter and toast cut-side down until golden.
Heat a cast-iron skillet until smoking hot. Place a beef ball on the skillet, smash it flat (¼ inch thick), and season with salt and pepper.
Cook 2 minutes until the edges are crisp, then flip, top with cheese, and cook 1 more minute. Repeat for the rest.
Spread creamy baconnaise on both buns, add the cheesy patty (or two for a double stack), and layer on lettuce, pickles, onions, or tomatoes. Close the bun and serve hot with fries or a cold drink.