Caramelized onions: Add 1 tablespoon of butter to a large pan over medium heat. 

When butter is bubbly and frothing, add all of your sliced onion. 

Add a pinch of salt and stir everything to combine. 

Reduce heat to medium-low and cook onions, stirring often for 20 minutes. 

After 20 minutes, your onions should be tan-colored, but not browned. You might start to see dark brown spots forming on the bottom of the pan though, and if you do see that, add a tablespoon of water or two to the pan with the onions and use that water to stir the dark parts at the bottom to keep things from burning. 

Continue cooking the onions over medium-low heat for another 10 or 15 minutes. Remember: if you see dark spots or dry areas on the pan, add a tablespoon of water. The water should cook off quickly but it will help to temper the pan temperature and help the onions to cook through without burning. 

After the onions are nice and brown, remove them from the pan to a plate to rest.

Special sauce: Add ssamjang and mayonnaise and stir together fully to make the ssamjang spread. 

Cheeseburger patty: Split your ground beef in half and form each patty into the shape of your bread. Make sure you form the patties larger than the bread because they will shrink when cooked. 

Wipe your pan out from the caramelized onions and add each patty when the pan is hot again. Cook over medium-high heat until they are done to your taste. This is around 3 to 4 minutes per side. 

When you have about 2 minutes left in your patty cooking process, add the cheese to the top of each patty and cook allowing the cheese to melt. For best melting, cover the pan with a lid or aluminum foil. This will steam the cheese and melt it.

Remove each patty to a plate while you start toasting the bread.

Build patty melt: spread a bit of butter on the bottom slices of bread and add each one to the pan. 

Add sauce to the exposed slice of bread in the pan. 

Top each piece of bread and sauce with a beef patty topped with cheese. 

Add onions on top of the melty cheese. 

Cook, toasting the bread for 2 or 3 minutes. 

Spread sauce on top slices of bread and add each slice to the top of the onions.

Add a little butter to the very top of the bread slices and carefully flip each sandwich. 

Cook bottom slices of bread for 2 to 3 minutes or until toasty and brown. 

Remove patty melts from pan, serve and enjoy.