Combine flour, egg, oil, dry yeast, salt, sugar, and warm water in a bowl. Knead for 10 minutes.
Proof the Dough
Shape into a ball, place in a bowl, cover, and let rise for 1 hour. (During this rising time, prepare the sauce, patties, and fries).
Make the Burger Sauce
Mix mayonnaise, ketchup, mustard, Tabasco, and parmesan cheese until smooth.
Cook the Beef Patties
Season patties with salt and pepper. Grill or sear for 1 minute per side (2 minutes total).
Fry the Potatoes
Fry potato fries in hot oil for 4 minutes until golden. Drain on paper towels.
Roll the Dough
Divide risen dough into 6 equal portions and roll each into a 10-inch circle.
Assemble the Stuffed Buns
Place a dough circle into a small bowl for support. Layer with 1 slice cheddar cheese, 1 beef patty, 2 tbsp sauce, pickled jalapeños, pickled cucumbers, second slice of cheddar, ⅙ of fries, and a third slice of cheddar.
Seal the Dough
Fold the dough edges over the filling and seal tightly. Place seam-side down on a baking sheet.
Bake
Brush with beaten egg and sprinkle with sesame seeds. Bake at 360°F (180°C) for 15 minutes.
Why shape the dough in a bowl? Placing the rolled dough inside a shallow bowl before layering your fillings creates a natural nest. This holds the heavy burger, sauce, pickles, and fries intact so you can tightly seal the dough edges without spilling any ingredients.