In a medium bowl, combine the ground beef, salt, and black pepper. Mix gently to combine without overworking the meat.
Divide the beef mixture into 4 equal portions and shape each into a thin patty slightly larger than the bread slices.
Heat olive oil in a large skillet over medium heat. Add the sliced onions and cook, stirring occasionally, until caramelized and golden brown, about 10 minutes. Remove onions from the skillet and set aside.
In the same skillet, cook the beef patties over medium heat for about 3-4 minutes per side, or until cooked through and browned. Remove patties from the skillet and keep warm.
Butter one side of each slice of rye bread.
Place 4 slices of bread, buttered side down, on a clean skillet or griddle over medium heat.
Layer each bread slice with one slice of cheddar cheese, one cooked beef patty, a generous amount of caramelized onions, and another slice of cheddar cheese.
Top each with the remaining bread slices, buttered side up.
Grill the sandwiches for about 3-4 minutes per side, pressing gently with a spatula, until the bread is golden brown and the cheese is melted.
Remove the sandwiches from the skillet, let rest for 1-2 minutes, then slice in half and serve immediately.