Very lightly toast the bread to firm it up, then set aside.
Divide ground chuck into three equal portions and flatten each into a large rectangle, 1-2" longer than the bread. (it will shrink during cooking)
Liberally salt and pepper each side of the meat.
In heated skillet, melt 2 tablespoons of butter and add onions. Saute until browned. Remove and set aside.
Return skillet to heat and add 2 tablespoons of butter then add hamburger patties, cook on medium-high, working in batches as needed. Flip once and cook until patties reach your desired degree of doneness.
While patties are cooking spread 1 tablespoon of butter to exterior of each piece of bread and spread the inside of slice for each sandwich with mayo. Place buttered side down and add a slice of cheese to each slice of bread.
Place the patty on non-mayoed slice of bread, top with onions, and cover with the mayoed bread.
Place patty melts in clean skillet on medium heat and let cook until the toasted bread turns golden brown and cheese has melted, flip and repeat.