In a medium bowl, combine ground beef, salt, black pepper, and garlic powder. Mix gently until just combined.
Divide the beef mixture into two equal portions and shape each into a flat patty roughly the size of the bread slices.
Heat a skillet over medium heat. Cook the patties for 4-5 minutes on each side or until fully cooked through and browned. Remove from skillet and set aside.
Spread butter evenly on one side of each slice of rye bread.
Place two slices of bread, buttered side down, on the skillet over medium heat.
Top each bread slice with two slices of cheddar cheese, then place a cooked beef patty on top of the cheese.
Add the remaining bread slices on top, buttered side up, to form sandwiches.
Cook the sandwiches for 3-4 minutes on each side or until the bread is golden brown and the cheese is melted, pressing gently with a spatula.
Remove from skillet, let rest for 1 minute, then slice in half and serve warm.