Step 1

Heat 2 tablespoons of the oil in a large (12-inch) skillet over medium. Add onions, and cook, stirring occasionally, until onions are tender and starting to brown, 8 to 10 minutes. Add sugar; reduce heat to low, and cook, stirring occasionally, until onions are browned and caramelized, 10 to 15 minutes. Add vinegar, and cook 30 seconds to deglaze the pan.

Step 2

Stir together dry mustard powder, salt and pepper in a small bowl. Place ground beef in a medium bowl, and sprinkle with dry mustard mixture. With your fingers, lightly work the mixture into beef, and shape into 4 (1-inch-thick) patties. Place patties on a plate, and freeze exactly 15 minutes.

Step 3

Heat a large (12-inch) cast-iron skillet over medium high, and add remaining 1½ tablespoons oil. From the freezer, place burgers directly into hot skillet. With a large metal spatula, firmly press each burger into skillet. Cook burgers 2 ½ to 3 minutes, without disturbing, so bottoms get browned and crusty. Flip burgers; top evenly with onions, and then sprinkle evenly with Gruyère. Place a lid on skillet, and cook burgers until cheese is melted and burgers are medium-rare, 1½ to 2 minutes. Place 1 burger on each roll, and serve hot with ketchup on the side.