Heat 2 tablespoons oil in a large (12-inch) sauté pan over medium heat, add the onions, and cook for 8 to 10 minutes, stirring occasionally, until the onions are tender and starting to brown. Add the sugar, reduce the heat to low, and cook for 10 to 15 minutes, stirring occasionally, until the onions are browned and caramelized. Add the vinegar and cook for 30 seconds to deglaze the pan.
Meanwhile, combine the dry mustard, 1½ teaspoons salt, and ½ teaspoon pepper in a small bowl. Place the ground beef in a medium bowl and sprinkle on the mustard mixture. With your fingers, lightly work the mustard into the beef and shape into four 1-inch-thick patties. Place them on a plate and freeze for exactly 15 minutes.
Heat a large (12-inch) cast-iron skillet over medium-high heat and add 1½ tablespoons oil. From the freezer, place the burgers directly in the hot skillet. With a large metal spatula, firmly press each burger into the pan. Cook the burgers for 2½ to 3 minutes, without moving them, so the bottoms get browned and crusty. Flip the burgers, then spoon on the onions and sprinkle the Gruyère on top. Place a lid on the skillet and cook the burgers for 1½ to 2 minutes, until the cheese is melted and the burgers are medium-rare inside. Place one burger on each roll and serve hot with ketchup on the side.
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