Place a medium skillet over medium heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add the onions to the pan and cook slowly, stirring occasionally, until they're deep golden brown, about 30 minutes. If the onions begin to stick to the pan, add a splash of water to loosen them up.
When the onions are almost done, use a large round cookie cutter to cut the crusts away from each slice of bread to make 8 bread discs. Set aside.
When the onions are ready, place a large skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons.
While the pan is heating up, combine the ground sirloin, parsley, Worcestershire, some salt and pepper in a medium-size mixing bowl. Divide the meat in four and shape each portion into thin patties that are about the same size as the bread discs. Cook the patties in the hot pan until golden brown and cooked through, about 5 minutes per side.
While the patties are cooking, butter one side of each round of bread. Remove the cooked patties from the pan, wipe the pan clean and place back over medium heat.
Assemble the beefcakes (with the buttered sides on the outside) layering some cheese, caramelized onions, and one patty per sandwich.
Place the stacks into the pan and cook until golden brown on both sides and the cheese has melted, 3-4 minutes per side.