Thinly slice ½ small yellow onion (about ½ cup). Cut 2 medium tomatoes into 8 slices. Core and cut ¼ medium head iceberg lettuce into fine shreds until you have 1 cup.

Place ¼ cup mayonnaise and ¼ cup ketchup in a small bowl and stir to combine. Split 4 soft sandwich rolls lengthwise. Spread the sauce onto the cut sides of the rolls (1 tablespoon per piece).

Place 1 pound lean ground beef, 1 ½ teaspoons adobo or sazón seasoning, and 1 teaspoon kosher salt (if using a no-salt seasoning) in a medium bowl. Gently mix with your hands until combined. Divide the mixture into 4 portions (4 ounces each). Form each portion into a patty about 3 ½-inches wide and ½-inch thick.

Heat a large cast iron skillet over medium-high heat. Add 2 of the patties and scatter half of the onions over the patties. Cook undisturbed until the bottom is beginning to brown, about 2 minutes. Flip the patties with a sturdy metal spatula or metal bench scraper. Use the spatula to chop the patty into small pieces.

Top each patty with 1 slice of the American cheese. Continue to chop and mix until the cheese is melted and the beef is cooked through, 2 to 3 minutes more.

Use the spatula to arrange the chopped beef into two portions about the size and shape of the rolls. Transfer the mixture to 2 of the rolls. Repeat cooking the remaining 2 patties.

Top the sandwiches with the lettuce and tomato slices. Carefully close each sandwich and transfer each one to a separate sheet of parchment paper or aluminum foil. Wrap each sandwich tightly in the parchment and cut in half.