Divide 1 pound ground beef into 6 equal, loosely packed rounds, about 2 1/2 to 3 ounces each. You’ll use 2 patties per stuffed bun, giving you 3 large buns total. Don’t pack the beef tightly.
Heat a stovetop griddle or large cast-iron skillet over medium-high heat and add the butter. Once hot, add the beef rounds, working in batches if necessary so they have plenty of room.
Immediately smash each round into a thin patty using a burger press or sturdy spatula. Season generously with salt. Cook undisturbed for about 2 minutes, until deeply browned with crisp edges. Flip and cook for another 1 to 2 minutes, or until cooked through. Transfer to a plate and allow to cool slightly.
Reduce the heat to medium and add the sliced onions directly to the beef drippings and butter left on the griddle. Cook, stirring occasionally, for 8 to 10 minutes, until soft, golden and caramelized. Add another small pat of butter if the pan becomes dry. Set aside to cool slightly.
Make the BBQ honey mustard by stirring together the mayonnaise, honey, mustard and BBQ sauce. Season with salt and pepper. In a separate bowl, combine the mayonnaise, sweet relish, ketchup and sugar for the burger sauce. Season with salt and pepper.
Preheat the oven to 400°F and line a sheet pan with parchment paper. Whisk the egg with 1 teaspoon water for the egg wash.
Unfold the cold puff pastry onto a lightly floured surface and gently roll just enough to smooth the seams. Divide the pastry into 3 equal pieces, making sure each piece is large enough to completely enclose 2 burger patties and the toppings.
Place one piece of puff pastry into a small bowl, allowing the excess pastry to hang over the sides. Layer in 1 slice cheddar cheese, 1 smash burger patty, a small amount of sauce, pickles and caramelized onions. Add the second burger patty and repeat the cheese, onions, pickles and sauce as desired. Don't overdo the sauce inside the pastry or it can become difficult to seal. Save plenty for dipping.
Bring the edges of the puff pastry over the filling and firmly pinch them together to completely seal the bun. Trim off any large amount of excess pastry so you don't end up with a thick, doughy bottom.
Carefully turn the bun out of the bowl and place it seam-side down on the prepared sheet pan. Repeat to make 3 large stuffed buns. If the puff pastry has become warm or very soft while assembling, refrigerate the buns for 10 to 15 minutes before baking.
Brush the tops and sides lightly with egg wash and sprinkle with white sesame seeds.
Bake at 400°F for about 20 minutes, or until the puff pastry is puffed, deeply golden brown and cooked through on the bottom. If the bottoms still look pale or doughy, bake for another 3 to 5 minutes.
Let the buns rest for 5 minutes before slicing. This gives all that melted cheese, sauce and burger juice a chance to settle inside. Slice in half and serve warm with both sauces on the side for dipping.