Measure 410 g all-purpose flour, 140 g mashed potato, 127 g brown sugar, 8 g salt, 8 g instant yeast, 190 ml water, 60 g butter, and 1 egg into a large mixing bowl.

Mix the bun ingredients until combined. Knead for about 8 to 10 minutes until the dough is smooth and slightly sticky.

Shape the dough into a tight ball. Cover and let it rise in a warm place for about 60 minutes, until doubled in volume.

Gently deflate the risen dough. Divide it into 8 portions of about 100 g each using as little additional flour as possible.

Shape each portion into a tight round bun. Arrange the buns on a parchment-lined baking sheet with space between them.

Cover the shaped buns and let them rise until doubled in volume, about 45 to 60 minutes. Preheat the oven to 180°C (356°F).

Beat the remaining egg for the egg wash. Brush the tops of the risen buns evenly with the beaten egg.

Bake the egg-washed buns at 180°C (356°F) for 12 minutes. Cool the buns on the baking sheet until safe to handle.

Combine 50 g homemade mayonnaise and 20 g ketchup in a small bowl. Stir until the base is uniform.

Add 7 g finely chopped pickles, 5 g finely chopped onion, 2.5 ml white vinegar, 2.5 g granulated sugar, and 0.5 g salt for sauce. Stir until the Thousand Island sauce is smooth and evenly combined.

Slice a cooled potato bun horizontally. Serve it with Thousand Island sauce as a burger bun or as a dipping sauce.