Mix all the ingredients for the burger sauce together and refrigerate until ready to use.
Divide the ground beef into 4 servings and form into burger patties.
Heat a large skillet over medium high heat and add ½ tbsp avocado oil into the pan.
Sprinkle a pinch of salt and pepper onto both sides of the patty and place in the pan. Cook the patties for about 3 minutes on each side for a medium cook. Cook longer if desired. After the patties are cooked, add 1 slice of cheese onto each patty, cover with a lid and let steam for about 30 seconds until melted. Remove from the heat.
Assemble the burger by placing one of the burger patties onto a couple iceberg lettuce leaves. Follow with a slice of tomato, sliced onions, and some of the burger sauce. Fold the lettuce over to create a secure lettuce wrap. Serve immediately.