In a medium bowl, using hands, combine beef, grated onion, egg, salt, pepper, and garlic powder. Form into six patties. Spread 1/2 teaspoon butter on one side of each bread slice; set aside. Heat a large heavy duty skillet over medium high heat. Cook burgers, turning once, until cooked through, about 8 mins. Transfer to a plate lined with paper towels. Drain fat from skillet. Add 3 bread slices, buttered side down, and layer each with 1 cheese slice, onion slice, burger, another cheese slice and bread, buttered side up. Cook over medium low heat, turning once, until bread is golden brown, about 4 mins. Repeat to make 3 more sandwiches.