Place a medium skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Add the onion to the pan and cook until tender, 4-5 minutes. Remove to a large mixing bowl and wipe the pan clean.
When the onions have cooled slightly, place the pan back over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. To the mixing bowl with the onions, add the sirloin, Worcestershire, some salt and pepper and mix thoroughly to combine. Divide the meat into 4 portions and shape into 4 thin burgers. Add the patties to the hot pan and cook until golden brown and cooked through, about 4 minutes per side.
While the burgers are cooking, butter up one side of each slice of bread then flip them over and spread mustard on each.