BOIL the potato in a medium pot until tender, 20 to 30 minutes. When cool enough to handle, peel the potato and mash. Measure 1/2 cup mashed potato and set aside (save the rest for another use).
WHISK together in the bowl of a stand mixer, or in a large bowl, the all-purpose flour, yeast and sugar. Set aside.
MELT 1/4 cup plus 1/2 tablespoon butter in a small saucepan. In a blender, combine the melted butter, milk and mashed potato, and blend until smooth.
ADD the potato mixture to the flour mixture and whisk to combine. Set aside for 30 minutes to give the yeast time to activate.
SWITCH to the dough hook attachment if using a stand mixer. Add the whole egg and one yolk, and mix until combined, then add the bread flour and 1 1/4 teaspoons salt, scraping down the bowl as needed until the ingredients are combined to form the dough.
IF MIXING by hand, beat in the egg and 1 yolk using a hand mixer, then add the bread flour and 1 1/4 teaspoons salt, mixing by hand with a spoon or spatula until the ingredients are combined to form a dough.
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.
TURN the dough out onto a lightly oiled work surface and portion the dough out into 1 1/2 dozen balls, each weighing about 1 1/4 ounces.
ROLL the portions into balls and place on a lightly oiled parchment-lined baking sheet (6 rows, 3 rolls each).
BRUSH the tops of the rolls very lightly 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.
HEAT the oven to 375° when the rolls are almost ready.
HEAT in a small saucepan, 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 3/4 teaspoon coarse sea salt.
BAKE the rolls until golden brown and completely set, about 20 minutes.
REMOVE and cool slightly before serving.
Nutrition information: Per each of 18 rolls: 135 calories, 3 grams protein, 18 grams carbohydrates, 1 gram fiber, 6 grams fat, 3 grams saturated fat, 43 milligrams cholesterol, 2 grams sugar, 234 milligrams sodium.