Place potato in a medium saucepan. Cover with water. Bring to a boil. Reduce heat to simmer and cook for 20 minutes or until potato is tender. Drain, reserving liquid. Add hot tap water to reserved liquid to measure 1 cup. Cool to 105 F. Mash the potato and set aside.
In the bowl of a stand mixer, whisk together 4 cups flour, sugar, salt and yeast. Add potato water, mashed potato, milk and butter all at once to flour mixture. Beat on medium speed until smooth. Using the dough hooks, knead in just enough additional flour one cup at a time, to form a smooth, soft, elastic dough. Transfer to a large, greased bowl. Turn dough over to grease all sides. Cover, place in a warm, draft free place and let rise for 35 minutes or until doubled in bulk.
Grease and lightly flour 2 baking sheets. Punch dough down. Divide dough into 24 pieces. Roll each piece into a ball and using the heel of your hand, flatten into a 5 inch round. Place flattened rounds on prepared baking sheets. Cover with a clean towel and let rise in a warm, draft free place for 20 minutes.
In a small bowl, whisk together egg and milk. After rolls have risen, brush with egg mixture and sprinkle lightly with sesame seeds.
Preheat oven to 400 F. Bake rolls for15 minutes. Remove from baking sheets and cool on wire rack. Slice with a serrated knife to serve with hamburgers.
Makes 2 dozen rolls.