Prick the potato all over with a fork. Then roast or microwave until completely soft, peel and mash with a fork. You'll need about 1/2 cup of this mashed potato
In the bowl of a stand mixer fitted with the dough hook attachment, combine flour, yeast, 1/2 cup mashed potatoes, butter and salt. Mix on low until the dough begins to come together, about 2 minutes. Then turn the speed up to medium and knead for about 6 minutes, until the dough is taught and can be pulled without tearing.
Transfer the dough to a separate bowl and tuck it into a smooth ball. Cover tightly and set aside to rise, 90 minutes.
Line a baking sheet with parchment paper. After the first rise, divide the dough into six equal pieces (about 100g each). Roll each of these pieces into a smooth ball and transfer to the parchment paper lined baking sheet. Cover the rolls with another baking sheet and set aside to rise again, about 90 minutes.
Preheat the oven to 400 F. After their second rise, brush each roll with egg wash and sprinkle with flaky salt. Bake until golden brown, 15-18 minutes. Allow to cool 5 minutes before slicing.