Combine mashed potatoes, milk, water, and sugar in a large mixing bowl or in the bowl of your stand mixer. Whisk until well-blended; sprinkle on the active dry yeast. Stir to combine and allow to proof for 5-10 minutes or until bubbles begin to form.

Stir in the salt and 3 cups of the flour

By hand: stir with a wooden or metal spoon until the dough comes together into a shaggy mass. Turn the dough out onto a floured work surface and knead for 5-10 minutes, adding flour as needed to keep it from sticking, until the dough is soft and the gluten developed. Add the remaining 1/2 cup of flour as necessary to create a pliable dough. With a mixer: Lock bowl into place and attach the dough hook. Turn the machine on to low and mix until most of the flour is absorbed. Add the remaining flour. When the dough pulls away from the side of the bowl, let the machine knead the dough for another 2-3 minutes.

If you've kneaded by hand, place the dough back in the mixing bowl. Cover the bowl with a tea towel, beeswax wrap, or plastic wrap. Allow to rise for one to one-and-a-half hours, until doubled in volume.

Preheat the oven to 375 degrees Fahrenheit. Transfer the dough to a lightly floured work surface and divide into 6 or 8 even pieces. Pick up each ball and use your hands to form the dough into smooth balls by folding the edges under as you turn the dough in your hands. This should leave a smooth surface at the top of the potato bun.

Gently move the dough balls to a baking sheet, allowing an inch or two between them. The egg wash can cause the edges of the buns to stick. If you like, place them on a baking sheet lined with parchment paper or a silicon baking sheet. Cover with a towel and leave to rise an additional 30-35 minutes or until doubled in bulk.

Whisk the egg in a small bowl and baste each bun. If desired, sprinkle on dried onion flakes or sesame seeds.

Bake for 15-20 minutes or until golden brown.

Use a spatula to life the buns to a cooling rack and allow to cool for at least 20 minutes before slicing.