Preheat oven to 425°F. Cut potatoes into thin fries. Rinse under cold water until water runs mostly clear. Pat completely dry with paper towels. Toss with olive oil, paprika, garlic powder, onion powder, salt, and pepper. Spread onto a parchment-lined baking sheet in a single layer. Bake for 45 to 55 minutes, flipping halfway, until deeply golden and crispy at the edges.
Prepare your Burger Sauce while the fries bake.
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.
Drop oven to 400°F once the fries are done or nearly done.
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.
