How to Make It
Stir together vinegar, water, sugar, and 1 teaspoon of the salt in a small saucepan over medium-high, and bring to a boil. Meanwhile, combine jalapeño slices and onion slices in a medium bowl. Pour boiling vinegar mixture over jalapeño and onion mixture. Let stand until room temperature, about 30 minutes.
Meanwhile, shape beef into 4 (3-ounce, 1/2-inch-thick) patties. Season evenly with remaining 1 teaspoon salt. Heat oil in a large cast-iron skillet over medium-high. Add patties, and cook until browned and crispy on both sides but still pink in the center, about 2 minutes per side. Transfer to a large plate, and set aside. (Patties will not be fully cooked.) Wipe skillet clean.
Place 4 bread slices on a clean work surface; top each with 1 cheese slice, 1 patty, and 2 tablespoons drained pickled jalapeños and onions. Top each with 1 cheese slice, and cover with remaining bread slices. Heat 1 tablespoon of the butter in cast-iron skillet over medium. Add 2 sandwiches to skillet, cover tops of sandwiches with a piece of aluminum foil, and place a heavy-bottomed pot or small cast-iron skillet on top, and gently press. Cook until golden brown and cheese has melted, 2 to 3 minutes per side, adding 1 tablespoon butter to skillet when flipping to the other side, and checking occasionally to prevent burning. Repeat with remaining sandwiches and 2 tablespoons butter. Serve patty melts immediately.