Special equipment: 2 (9-inch) round or square cake pans
Cook the potato in a microwave on HIGH until soft and it squeezes easily, using the designated baked potato setting, or up to 15 minutes. Peel and pass the warm potato through a potato ricer or food mill into a large bowl. (You should have 1 cup pureed potato.)
Stir the hot water, eggs, half the sugar and yeast into the potato. Add 2 cups of the flour and the salt and mix with a wooden spoon to make a sticky, shaggy dough. Cover the bowl tightly with plastic wrap and set aside in a warm place until the dough doubles in volume, about 1 1/2 hours.
Beat the butter with the remaining 1/3 cup sugar in a standing mixer fitted with the paddle attachment until light and fluffy. Switch to the dough hook and add the risen dough to the creamed butter. Continue to mix on low until the butter and dough come together, about 1 minute. (Stop the mixer and scrape down the bowl if needed; the dough will be very sticky.) Gradually add the remaining 2 1/4 cups of flour, about 1/4 cup at a time, to make a shaggy dough that pulls away from the side of the bowl. Continue kneading on medium speed until dough is smooth but still tacky, about 3 to 4 minutes.
Turn dough onto a lightly floured work surface and knead by hand until dough is smooth and no longer tacky, 1 to 2 minutes more. (If the dough is still sticky, gradually add 1/4 cup flour.) Shape dough into a ball.
Brush a large bowl with butter and turn dough around in bowl to coat lightly. Cover bowl tightly with plastic wrap, trace a circle the size of the dough on the plastic, and note the time. Let rise at room temperature until doubled in size, about 1 1/2 hours.
Generously butter two 9-inch round or square cake pans. Turn the dough out of the bowl and pat into a rectangle about 16- by 8-inches, gently pressing out excess air. Divide the dough into 32 equal portions, about an ounce each, with a pizza wheel or bench scraper.
(If you don't have a scale, divide the dough in half lengthwise, then in half crosswise. Cut each of those four sections into 8 equal-sized rolls.)
Tuck the edges of the dough under to make round rolls and place them seam-side down in the prepared pan, leaving a little space in between each roll. (See Cook's Note) Cover the pan with buttered plastic wrap and set aside in a warm place until the rolls rise almost to the rim of the pan and have more than doubled in size, about 45 minutes.
Position a rack in the center of the oven and preheat to 375 degrees F.
Bake rolls until golden brown and puffy, and an instant-read thermometer inserted into the center of the rolls registers 190 degrees F, about 40 minutes. Remove rolls from the oven and quickly brush the tops with soft butter.
Cook the rolls in the pan for about 10 minutes before turning them out onto a rack in one piece. Cool slightly. Serve warm or at room temperature in one piece or pulled apart as individual rolls.
At the point the rolls are formed, placed in the pan and covered, they can also be refrigerated overnight. If rolls have been refrigerated overnight, the final proofing time will be closer to 1 1/2 hours. Storing Tips: Storing and serving rolls in one piece keeps them moist and soft. Store baked and completely cooled rolls in one piece in an air-tight plastic bag, or wrapped loosely in plastic wrap, then foil.
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