Step 1

Cover the potatoes with water in a medium saucepan, bring to a boil over high heat, then reduce the heat to medium-low; simmer until the potatoes are cooked through, about 8-10 minutes. Transfer 5 tablespoons of the potato water to a bowl then drain the potatoes and return them to the hot saucepan. Over low heat, cook the potatoes while shaking to remove the surface moisture, about 1 minute.

Step 2

Process the potatoes with a food mill or ricer (or mash well with a masher - they should be very smooth) into the saucepan. Measure out 1 cup of potatoes, add to a medium bowl and mix in the butter until melted. Throw a kitchen towel or some foil over the bowl to keep the potatoes warm.

Step 3

In the bowl of a stand mixer, combine the flour, sugar, yeast, and salt. Add the warm potatoes and mix with your hands until combined. Mix in 1 egg and the reserved potato water with the dough hook attachment and continue to mix on low speed until the dough is soft and a little sticky, about 8-10 minutes. The dough should still be warm.

Step 4

Remove the dough and shape into a ball. Lightly grease the mixer bowl and return the dough to the bowl, coating it lightly in the oil. Cover tightly with plastic wrap and let the dough rise in a warm spot until almost doubled in volume, about 30-40 minutes. Very warm kitchens will require less time.

Step 5

Turn the dough out onto a lightly floured surface, pat it into an 8-inch square, and cut the square into 9 equal pieces; separate the pieces and cover them loosely with plastic wrap to prevent a skin from forming on the top. Working with one at a time, roll the dough rounds into a tight balls and return them to the work surface under the plastic wrap; allow them to rest for 15 minutes.

Step 6

Line two large baking sheets with parchment paper. Firmly press each dough round into a 3 ½-inch disk, pushing out large pockets of air from the dough. Line the disks up on the baking sheets and cover loosely with plastic wrap. Allow the dough to proof at room temperature until almost doubled in size, another 30-40 minutes. While the dough proofs, preheat the oven to 425° F with the racks in the two center positions of the oven.

Step 7

In a small bowl, mix the remaining egg with a tablespoon of water. Lightly brush the tops of the dough rounds with the egg wash and sprinkle with sesame seeds, if using.

Step 8

Bake for 15-18 minutes, rotating pans halfway through the time from top to bottom and front to back, until the buns are a deep golden brown. Transfer the pans to wire racks to cool for 5 minutes then transfer the buns to the racks to cool completely before slicing.