Heat a medium skillet over medium heat.
Melt 2 tablespoons of butter in the skillet and add the diced onions. Cook, stirring occasionally, for about 15 minutes.
When the onions are soft and beginning to brown, add 1/4 cup of water and continue cooking, stirring occasionally, until the water has evaporated. Repeat with the remaining 1/4 cup of water.
After about 30 minutes of cooking the onions should be golden brown, soft, melty, and pure perfection.
Sprinkle with salt. Set aside.
While the onions are cooking prepare the secret sauce. Stir together the ketchup, mayo, vinegar, and relish. Set aside.
Heat a griddle or large cast iron skillet to medium.
Melt the remaining butter on the hot griddle and place the buns, cut side down, on the griddle. Cook until golden brown.
Spread around 1 tablespoon of the sauce on the bottom of each bun. Set aside.
Turn the heat to high. While the griddle is heating, form the meat into 8 small balls of meat, loosely compacted.
Place the balls of meat a few inches apart on the griddle and sprinkle the tops with salt. Let cook one minute.
Flip the balls over and immediately take a small plate (or flat spatula that doesn't have holes for the meat to squish through) and press the meat flat, forming a thin patty the same diameter as your bun.
Squirt about one teaspoon of mustard on each patty. Cook for 1-2 minutes and flip again.
Immediately place the cheese on each patty and top half of the patties with a heaping spoonful of the onions.
Cook 1 minute or until patties are cooked through and cheese is beginning to melt.
Use a spatula to pick up one of the patties without the onions and place it on top of a patty with the onions.
Let cook for 30 more seconds to meld everything together.
Remove from griddle to the grilled buns. Top with lettuce, tomato, and pickles, if desired.