Preheat oven to 400°F.
Heat a skillet over medium-high heat. Add burger patties and let them brown before seasoning. Once browning starts, season with salt and pepper. Cook 2 to 3 minutes per side until cooked through. Set aside to cool slightly.
Dip 2 rice paper sheets one at a time into the beaten egg wash until slightly softened. Lay them overlapping over a small round bowl or ramekin. Layer in this exact order: one burger patty, one slice of cheddar, 1 tbsp burger sauce, 3 pickle slices, 1 tbsp diced raw onion, second burger patty, second slice of cheddar. Fold the rice paper edges up and inward, sealing tightly. Flip onto a parchment-lined baking sheet seam-side down. Brush the top and sides generously with remaining egg wash. Sprinkle with sesame seeds. Repeat with the second Wellington.
Bake at 400°F for 18 to 22 minutes until deeply golden, crispy, and browned all over.
Serve hot with crispy fries and extra burger sauce on the side.
