Melt butter in a small skillet over medium-low heat. Add onions and cook, stirring occasionally, until onions are a deep golden color, about 20-25 minutes.
For The Burgers
Divide beef into four 4-ounce portions. Roll into balls.
Preheat a griddle or large cast-iron skillet over high heat. Add the burgers to the skillet one at a time and press down with a burger press or heavy spatula until flattened to about ¼-inch (use a square of parchment between the burger press and patty to avoid sticking). Season the tops with salt and pepper.
Cook for 2-3 minutes or until edges are browned. Flip burgers. Add a cheese slice to the top of each burger and cook for 2 minutes more or until desired doneness.
To serve, make a stack of 4 or 5 lettuce leaves. Add burger patty and top with burger sauce, caramelized onions and toppings as desired. Wrap lettuce around the burger.
Notes
To make a low-carb burger sauce, combine:
¼ cup mayonnaise
1 tablespoon sugar-free ketchup
1 tablespoon chopped pickles
1 teaspoon Dijon mustard
1 teaspoon pickle juice
⅛ teaspoon paprika
Keep leftovers in an airtight container in the refrigerator for up to 4 days.