Caramelize Onions: Heat butter and olive oil on a griddle or skillet over medium heat. Add the sliced onions with a pinch of salt. Cook for 10–15 minutes, stirring occasionally, until golden and soft. Stir in Worcestershire sauce and balsamic glaze, cooking another 2–3 minutes. Remove from heat and set aside.
Make Burger Sauce: In a small bowl, whisk together mayonnaise, ketchup, mustard, dill pickle juice, paprika, and hot sauce if using. Refrigerate until ready to assemble.
Cook Burgers: Preheat griddle to 450–500°F. Place the ground beef balls on the hot surface and smash them immediately with a flat metal spatula using firm, even pressure. Season with salt and pepper. Cook for 1½–2 minutes until edges are crispy.
Add Cheese: Flip patties, place a slice of cheese on each, and cook for an additional 1 minute until melted.
Toast Buns: On the same griddle, toast the cut sides of the buns until golden.
Assemble Burgers: Place pickle slices on the bottom bun, top with the cheesy patty, add caramelized onions, and spread burger sauce on the top bun. Serve immediately.