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 cheddar 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, ensuring a clear border around the cheese.
Top with the remaining patties and pinch the edges to seal the cheese inside.
Heat 1 tablespoon of olive oil over medium-high heat. Add the onion and salt, cooking until golden brown, 9 to 10 minutes. Transfer to a bowl.
Reduce heat to medium, add the remaining olive oil, and stir in the garlic, cooking for about 45 seconds.
Add the Swiss chard, salt, and pepper. Cook until wilted, about 4 to 5 minutes. Transfer to a bowl.
Wipe the skillet, return to heat, and add the patties. Cook until browned and cooked through, about 9 to 10 minutes.
Place the bottom half of the rolls on serving plates. Top with lettuce, a slice of tomato, a pattie, and onions. Close the sandwiches.
Let cool for 2 to 3 minutes before serving. Meanwhile, heat the Swiss chard in the skillet for 1 minute before serving with the burgers.
