Caramelize the onions: Heat butter and olive oil in a pan over medium low heat. Add sliced onions with a pinch of salt and cook slowly, stirring occasionally, for about 25 to 35 minutes until deeply golden and jammy. Add a small splash of Worcestershire if using. Set aside and let cool completely.
Make the burger sauce: In a bowl, mix mayo, ketchup, mustard, relish, garlic powder, paprika, and white vinegar. Stir until smooth and refrigerate. Reserve half for dipping later.
Cook the smashed patties: Heat a cast iron skillet over high heat until very hot. Add the beef balls and smash them thin. Season with salt and pepper. Cook for 1 to 2 minutes until crispy, flip and cook another 1-2 minutes. They will cook more in the oven so no need to over cook here. Remove from heat.
Prep the pastry: Lightly flour a surface and roll the puff pastry slightly larger, about 10 to 12 inches. Place it on a parchment-lined baking sheet. Use a bowl to cut a circle out of the puff pastry. Depending on how big your patties are, you’ll want a big enough circle to wrap around. Mine was around 8 inches.
Assemble: Add burger patty, 2 slices of American cheese (I cut my slices so they were a little smaller), Add some onions, a dollop of sauce, another patty, and another 2 slices of cheese.
Wrap: Fold the puff pastry up and over the burger, sealing all edges tightly. Flip so the seam is on the bottom.
Finish: Brush the entire pastry with egg wash and sprinkle generously with sesame seeds.
Bake: Bake at 425°F for 15 to 20 minutes until the pastry is deeply golden brown and puffed.
Rest and serve: Let rest for 5 to 7 minutes before slicing. Serve warm with the reserved burger sauce on the side.
