Step 1Boil the potato in a medium pot until tender, 20 to 30 minutes. When cool enough to handle, peel the potato and mash. Measure one-half cup (2 2/3 ounces) mashed potato and set aside (save the rest for another use).
Step 2In the bowl of a stand mixer, or in a large bowl, whisk together the all-purpose flour, yeast and sugar. Set aside
Step 3In a small saucepan, melt one-fourth cup plus one-half tablespoon butter. In a blender, combine the melted butter, milk and mashed potato, and blend until smooth.
Step 4Add the potato mixture to the flour mixture and whisk to combine. Set aside for 30 minutes to give the yeast time to activate.
Step 5If using a stand mixer, switch to the dough hook attachment. Add the whole egg and one yolk, and mix until combined, then add the bread flour and 1¼ teaspoons salt, scraping down the bowl as needed until the ingredients are combined to form the dough.
Step 6If mixing by hand, beat in the egg and 1 yolk using a hand mixer, then add the bread flour and 1¼ teaspoons salt, mixing by hand with a spoon or spatula until the ingredients are combined to form a dough.
Step 76. Scrape the dough into a large, lightly oiled bowl, and lightly oil the top of the dough, then cover the top of the bowl with plastic wrap. Set aside until it doubles in size, about 30 minutes.
Step 8Turn the dough out onto a lightly oiled work surface and portion the dough out into 1½ dozen balls, each weighing about 1¼ ounces.
Step 9Roll the portions into balls and place on a lightly oiled parchment-lined baking sheet (6 rows, 3 rolls each).
Step 10Very lightly brush the tops of the rolls with oil and gently cover with a layer of plastic wrap. Set aside in a warm place until almost doubled in size, about 1 hour.
Step 11When the rolls are almost ready, heat the oven to 375 degrees.
Step 12In a small saucepan, heat the remaining 2 tablespoons butter over low heat just until melted. Place the butter in a small bowl and beat in the remaining egg yolk. Gently brush the tops of the risen rolls with the butter mixture, then sprinkle over the remaining three-fourths teaspoon coarse sea salt.
Step 13Bake the rolls until golden brown and completely set, about 20 minutes.
Step 14Remove and cool slightly before serving.