In a small bowl, whisk together ½ cup mayonnaise with 1 Tbsp Yucatán sauce and the juice of half of a lime. Season to taste with salt and pepper and set aside until ready to serve. \nIn a small bowl, mash the avocado with remaining lime juice and a pinch of salt. Set aside. \nIn a large bowl, mix together the ground beef, cilantro, breadcrumbs and remaining ¼ cup Yucatán sauce. Form the meat into four equal-sized patties, about 1-inch think. Press your thumb into the top of each patty, making an indent. (This will help to ensure the meat cooks evenly). Season both sides with salt and pepper.\nBrush your onion rounds with olive oil and sprinkle with a pinch of salt. \nHeat a grill or grill-pan to medium-high heat. Place burgers and onion rings on the grill. Grill burgers until lightly charred on the bottom side, about 3-4 minutes. Flip over, top with cheese. Continue grilling until cooked through to desired doneness, about 3-5 minutes more. Grill onions until softened and well charred on both sides, about 5-7 minutes per side. Remove to a cutting board or serving plate. \nWhen your burgers are almost finished cooking, place buns, cut side down, on the grill and grill until toasted. About 2 minutes. \nSpread Yucatán mayonnaise over both sides of toasted buns. Place burger on the bottom bun and top with grilled onions. Spread a large smear of avocado mash on the top bun. Sandwich together and enjoy!