Prepare the Onions: To a large skillet over medium heat, add oil, swirling the pan to coat. Add the thinly sliced onions. Let them cook for about 5 minutes while you prepare the hamburger patties.
Make the Patties: In a large mixing bowl, combine the ground beef, hamburger seasoning, seasoned salt, and garlic powder. Mix until just combined. Divide the ground beef mixture into four equal portions (4 ounces each). Shape each portion into a thin patty, about 5 inches in diameter, or slightly larger than the hamburger buns to allow for shrinkage during cooking.
Cook the Patties: Arrange the cooked onions into 4 piles. Top each pile of onions with a ground beef patty. Using a spatula, press and flatten the patties into the onions. Cook the patties for about 4-5 minutes per side, or until desired doneness is reached, carefully flipping the patties with the onions. For medium-done burgers, the internal temperature should be 145°F-150°F. For medium-well, cook to 150°F-165°F.
Melt the Cheese: Just before removing the burgers from the heat, place a slice of cheddar cheese on each patty to melt.
Toast the Buns: While the burgers are cooking, lightly toast the hamburger buns in a separate pan or in the oven.
Assemble the Burgers: Spread a generous amount of fry sauce on the bottom halves of the hamburger buns. Top each with 2 cheeseburger patties. Add optional toppings such as lettuce, tomato, pickles, ketchup, and mustard for garnish.