Heat a large cast-iron skillet over medium-high heat. Add the butter. When it foams, add the bread, cut-side down, and toast until golden brown, about 2 minutes. Transfer the bread to a platter.
Sprinkle the beef with salt and pepper and mix it in, using your hands. Shape the ground beef into four patties. Flatten the patties between sheets of wax paper to a 3/8-inch thickness. (The patties will be 1 1/2 times larger than your palms; it's ok if the edges look ragged, since the burgers shrink up a bit while cooking.)
Wipe the skillet lightly with vegetable oil using a paper towel. Add the patties to the skillet and cook on the first side for 3 minutes. Flip, season again, and cook for 2 minutes. Top each patty with a slice of American cheese and cook 1 minute more.
Smear the toasted buns on both sides with some of Nancy's Special Sauce. Stack a burger patty on top of each bun, add a slice of onion, then top with a second patty, followed by the pickles, tomatoes and lettuce. Cover with the top buns and eat with a napkin!
In a bowl, stir together the mayonnaise, ketchup, mustard, relish, Sriracha and vinegar until well-combined.