Caramelize the Onions: Peel and slice the onions into thin half-moons or dice them to about ½ inch. Place a large skillet (at least 12 inches) over medium-low to medium heat. Once the pan is warmed, add the oil, followed by the prepared onions. Cook, stirring occasionally, for 25-35 minutes, or until they achieve a rich, sweet flavor and deep golden-brown color. Season with sea salt to your liking during the last few minutes of cooking. Remove from heat and set aside. This should yield approximately 1 ½ cups of caramelized onions.
Prepare Burger Components & Sauce: While the onions are slowly caramelizing, prepare the remaining ingredients. Wash and dry the lettuce leaves. Slice the tomatoes and pickles. In a small bowl, thoroughly combine all ingredients for the Special Burger Sauce: avocado oil mayonnaise, sugar-free ketchup, sugar-free pickle relish, and white wine vinegar. Set aside.
Form and Season the Patties: Gently shape the ground beef into 6 equal patties, each about ¾-inch thick. To avoid tough burgers, handle the meat as little as possible. In a small dish, mix the yellow mustard, smoked paprika, and garlic powder. Evenly spread this mustard mixture on one side of each patty. Season both sides of the patties with sea salt (if using) and freshly ground black pepper.
Cook the Beef Patties: Preheat your grill or a heavy-bottomed skillet to a consistent medium-high heat, approximately 375-425°F (190-220°C). Place the patties mustard-side down onto the hot surface. Cook for 5-9 minutes on the first side. Flip the patties carefully. If using cheese, place a slice on top of each patty immediately after flipping. Continue to cook for another 5-9 minutes on the second side, or until the burgers reach your desired internal temperature and level of doneness. (A minimum internal temperature of 160°F (71°C) for ground beef is recommended for safety).
Assemble the Wraps: Once the patties are cooked, assemble your Cali-Fresh Burger Lettuce Wraps without delay. Lay out two lettuce leaves per serving to form a sturdy "bun." Place a cooked beef patty (with cheese, if used) onto the lettuce. Top generously with a portion of the caramelized onions, sliced tomatoes, and dill pickles. Drizzle or spread with the Special Burger Sauce.
Serve Immediately: Enjoy your flavorful, low-carb burger wraps right away.