Melt 1 T. butter in a skillet over medium high heat. Cook 2 patties until well browned about 5 mins. Remove from pan and cook remaining patties.
Pour off the grease. Add 1 t. oil, 1 T. butter, onions and 1/2 t. salt and cook occasionally, until golden brown,8-10 mins.Place patties,browned side up,on top of the onions. Reduce heat to medium and cook until onions are tender and patties are cooked through,about 5 mins.
Divide 1 c cheese among 4 slices of bread. Top with patties,onions and remaining cheese and bread. Wipe out skillet with paper towels. Melt 1 T. butter in same skillet over medium heat.Cook 2 sandwiches until golden brown and cheese is melted,about 3-4 mins. per side. Transfer to a warm plate and cook remaining sandwiches in 1 T. butter.