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Heat the oil in a large skillet over medium-high heat. Alternately, heat a grill to medium (350 degrees F) and rub the grate with a towel dipped in oil.
Heat the broiler on low heat.
Add the patties to the skillet, and cook for about 4 to 6 minutes. Flip the burgers and cook another 4 to 6 minutes for medium-rare burgers. Alternately, grill the patties uncovered for 4 to 6 minutes, then flip and grill for another 4 to 6 minutes. Top each patty with a piece of cheese and put under the broiler to melt the cheese. Alternately, close the grill top, and cook until the cheese is melted.
For the each burger, spread the bottom of the roll with 1/4 of the mayonnaise. Put the burger on the mayonnaise and top with 2 slices of the bacon and some of the onions. Cover with the top of the roll and serve.