To make the sauce, stir together the mayo, mustard, ketchup, pickle brine, cayenne, and salt in a small bowl until well combined.

Heat a cast-iron skillet over medium heat for two minutes.

Open the hamburger buns and spread each inside face with some of the softened butter.

Place buns buttered side down (probably in two batches, depending on your skillet) and toast for a couple of minutes, until light golden brown on the underside.

Place buns on serving plates.

Raise heat to medium-high.

Divide beef into four hockey pucks and sprinkle both sides generously with salt and pepper.

Place pucks onto skillet and use a sturdy metal spatula to firmly smash each puck into a patty that's about ⅓-inch thick. After this, absolutely ignore the burgers for about three minutes, until the undersides are brown and crisp at the edges and the juices on the surface are hot and bubbly.

Scrape up each patty and any brown bits with the spatula and flip it.

Cook one to two minutes more.

Place a slice of cheese on top of each burger at the very end of cooking to melt just enough.

Transfer patties to buns.

Layer each burger with two tomato slices and a piece of lettuce.

Spoon ¼ of the sauce onto each top bun half. Serve immediately.