Fire up your grill with a griddle or grill pan over direct medium-high heat.
Clean your griddle well. I use a brush and water, it works better than a scraper. Just be careful adding water to a hot griddle. DO NOT have any body parts nears the griddle when you add the water as it will instantly create a cloud of very hot steam. Then scrub, scrub, scrub. Repeat as necessary. You want the griddle clean to toast the bread.
Lightly spread a thin layer of mayonnaise on one side of each slice of bread. Working in batches if necessary, lay the bread mayo-side down on the hot griddle and toast for 2-3 minutes or until lightly brown and crunchy.
Flip and lightly brown the non-mayo sides too. Remove to a plate for the final assembly step.
Once the griddle is hot add 2 tablespoons of butter. Once melted add the onion and cook for 20 minutes or longer, until they are golden brown and caramelized. Stir and flip often with a spatula or tongs. If they start to burn reduce the heat. If they get too dry add a bit more butter.
Stir in the Worcestershire sauce and cook for 1 more minute. Remove to a bowl and cover with foil or a plate to keep warm.
Form burger into 4 5-ounce patties. Don't pack them tightly, just get them lightly shaped.
Season beef on both sides. Don't be shy with it.
Add remaining 2 tablespoons of butter to the griddle. Add the burgers. Let them cook for 3 minutes then smash out as thinly as possible using a burger press.
Continue cooking burgers until you see a bit of a crunchy edge. Flip.
Cook burgers another 3 minutes. Add 1 slice of Swiss and 1 slice of pepper jack to each patty.
Divide the onion between the patties. Grill another minute or two to melt the cheese.
Divide sauce between the non-mayonnaise sides of the bread. Use a little or a lot, it's up to you.
Add the cooked patties and top bread slices and serve!
Notes
This recipe is best made using a griddle on a gas or charcoal grill. You can also use a griddle grill. If you don't have a griddle you can use a large cast iron skillet, but you'll have to cook the burgers in batches.