In a medium bowl, gently mix the beef salt and pepper. Form the mixture into 8 thin patties 3½ to 4-inches wide. Stack the cheese and cut into a total of 16 equal-size pieces. Place 4 of the patties on your work surface and top each with 4 pieces of the Cheddar, making sure you leave a border clear of cheese all the way around. Top with the remaining patties and firmly pinch the edges together to seal in the cheese.
SAUTE THE ONIONS:
In a large skillet, heat 1 tablespoon of the oil over medium-high heat. Add the onion and salt and cook, stirring occasionally, until tender and golden brown, 9 to 10 minutes. Transfer to a bowl.
COOK THE SWISS CHARD:
Reduce heat on the skillet to medium and add the oil. Stir in the garlic and cook until it just begins to brown around the edges, about 45 seconds. Add the Swiss chard, salt and pepper. Cook, tossing occasionally, until the chard is wilted and tender, 4 to 5 minutes; transfer to a bowl.
ASSEMBLE THE BURGERS:
Wipe the skillet with a paper towel to remove any moisture, return it to medium heat and add the patties. Cook, turning once, until they are browned and cooked through, 9 to 10 minutes.
Place the bottom half of each roll on a serving plate. Top each with a lettuce leaf, tomato slice, pattie, and some of the onion; close the sandwiches. Let the burgers cool for 2 to 3 minutes before serving as the cheese will be very hot.
Meanwhile, heat the skillet over medium-high heat. Return the chard to the pan and cook to heat through, about 1 minute. Serve the greens with the burgers.
Notes
If making this recipe as EGG FREE, use buns that have no egg in the ingredients. If making this recipe as GLUTEN FREE, use gluten free buns.