Divide the ground beef into four 4-ounce portions and roll them loosely into balls. Keep cold until ready to sear.
In a small mixing bowl, whisk together the mayonnaise, mustard, ketchup, and pickle brine to create the burger sauce. Set aside.
Heat a large cast-iron skillet over high heat until it begins to smoke slightly.
Place the beef balls into the dry skillet, leaving ample space between them. Using a heavy-duty spatula or a burger press, flatten each ball into a very thin patty (approximately 1/4 inch thick).
Season the patties generously with salt and pepper. Sear for 2 minutes without moving them to develop a deep brown crust.
Flip the patties using a sharp metal spatula, scraping to ensure the crust remains attached. Place a half-slice of cheese on each patty and cook for an additional 60 seconds.
Divide the shredded lettuce, tomatoes, red onions, and pickles between two large bowls.
Place two smash burger patties into each bowl and drizzle with the prepared burger sauce immediately before serving.