Special sandwich sauce: combine all ingredients in a small mixing bowl and whisk to combine. 

Add very finely diced pickles if you want a little texture and/or add cayenne powder if you want spice/heat in your sauce. 

Onion slicing: using a Japanese or other type of mandoline slicer, slice 1/4 of an onion into very thin slices. You want the slices to be almost paper-thin.

Cheeseburger patty: weigh your ground beef into two 2.5 to 3-ounce portions and then shape them into balls. If you do not own a scale, you should be able to get six 2.5-ounce-ish balls out of one pound of ground beef. 

Preheat a griddle or pan to at least 400 F (205 C) for around 5 to 10 minutes. 

Toasting the bread slices: Add a bit of butter to both slices of bread and toast them on the hot griddle for about 1 minute. Add butter to the second side of each piece of bread—flip and cook for another 1 minute. Continue flipping and cooking the toast until both sides are golden brown. Remove the toast to a cooling rack to rest while you cook the burgers.

Burger cooking: add the beef balls to the hot surface, top each ball with half of the paper-thin slices of onion, and smash everything down as flat as you can on the griddle or pan. You're going to want to use a lot of pressure and smash hard. The goal here is to get as much surface area of the meat to touch the hot surface as possible. If you have a flat top or a griddle, you might want to use some sort of very heavy spatula or even a clean finishing trowel from your local hardware store.

As soon as you've finished smashing, season the top of the onions with a pinch of salt and black pepper. You will season again when you flip, but this is when you should do most of the seasoning. 

Cook for 2 minutes on this first side or until the very edges of the onions that are touching the pan and the patties start to turn brown.

Use a sharp spatula and scrape under each patty to flip. The goal here is to make sure you don't lose any of the crispy brown bits you worked hard to build up with the smashing process. 

Once flipped the patty will be resting on top of a bed of onions, season the top of the beef with a small pinch of salt and pepper and add one slice of American cheese on top of each patty. 

Cook the patties on top of the onions for 3 to 4 minutes or until the onions under the patty are browning well. After 3 to 4 minutes, using a spatula, scrape up and lift one patty off of the griddle or pan and stack it on the other cheese-topped patty. Carefully remove both patties to a plate to prepare the patty melt construction.

Burger construction:

Spread some special sandwich sauce on the bottom slice of bread and top with the double fried-onion patties. Add pickles (if desired) and spread a bit more special sauce on the second slice of bread before closing the patty melt. Serve and enjoy.