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Equipment: 2 (9-inch) round or square cake pans
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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.
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Peel and pass the warm potato through a potato ricer or food mill into a large bowl.
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(You should have 1 cup pureed potato.)
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Stir the hot water, eggs, half the sugar and yeast into the potato.
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Add 2 cups of the flour and the salt and mix with a wooden spoon to make a sticky, shaggy dough.
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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.
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Beat the butter with the remaining 1/3 cup sugar in a standing mixer fitted with the paddle attachment until light and fluffy.
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Switch to the dough hook and add the risen dough to the creamed butter.
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Continue to mix on low until the butter and dough come together, about 1 minute.
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(Stop the mixer and scrape down the bowl if needed; the dough will be very sticky.)
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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.
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Continue kneading on medium speed until dough is smooth but still tacky, about 3 to 4 minutes.
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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.
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(If the dough is still sticky, gradually add 1/4 cup flour.)
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Shape dough into a ball.
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Brush a large bowl with butter and turn dough around in bowl to coat lightly.
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Cover bowl tightly with plastic wrap, trace a circle the size of the dough on the plastic, and note the time.
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Let rise at room temperature until doubled in size, about 1 1/2 hours.
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Generously butter two 9-inch round or square cake pans.
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Turn the dough out of the bowl and pat into a rectangle about 16- by 8-inches, gently pressing out excess air.
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Divide the dough into 32 equal portions, about an ounce each, with a pizza wheel or bench scraper.
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(If you don't have a scale, divide the dough in half lengthwise, then in half crosswise.
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Cut each of those four sections into 8 equal-sized rolls.)
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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.
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(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.
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Position a rack in the center of the oven and preheat to 375 degrees F.
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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.
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Remove rolls from the oven and quickly brush the tops with soft butter.
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Cool the rolls in the pan for about 10 minutes before turning them out onto a rack in one piece.
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Cool slightly.
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Serve warm or at room temperature in one piece or pulled apart as individual rolls.
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Cook's Note: At the point the rolls are formed, placed in the pan and covered, they can also be refrigerated overnight.
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If rolls have been refrigerated overnight, the final proofing time will be closer to 1 1/2 hours.
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Storing Tips: Storing and serving rolls in one piece keeps them moist and soft.
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Store baked and completely cooled rolls in one piece in an air-tight plastic bag, or wrapped loosely in plastic wrap, then foil.
