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1
Generously cover potato with cold water in a medium saucepan. Bring to a boil, then simmer, covered, until very tender, about 10 minutes. Reserve 1 cup cooking liquid, then drain potato well.
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2
Meanwhile, melt 2 1/2 tablespoons butter.
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3
Mash hot potato in a large bowl with a fork. Stir in milk, salt, 2 tablespoons sugar, and 2 tablespoons melted butter (mixture will be lumpy).
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4
Cool 1/2 cup cooking liquid to warm (105 to 115u00b0F). Stir in yeast and let stand until foamy, about 5 minutes. (If mixture doesn't foam, Start over with new yeast and remaining cooking liquid.)
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5
Stir yeast mixture into potato mixture, then stir in flour with a wooden spoon until a soft dough forms.
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6
Turn out dough onto a floured surface and knead, dusting surface and hands with just enough flour to keep dough from sticking, until smooth and elastic, about 10 minutes (dough will be slightly sticky).
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7
Brush a large bowl with some of remaining melted butter, then turn dough in bowl to coat. Cover tightly with plastic wrap and let rise, chilled, 8 to 12 hours.
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8
Punch down dough (do not knead), then halve. Roll each half into a 12-inch-long log on a very lightly floured surface with lightly floured hands. Cut each log into 12 equal pieces and roll each into a ball. Arrange evenly spaced in 6 rows of 4 (less than 1/2 inch apart) in a buttered 13- by 9- by 2-inch baking pan. Cover pan with a kitchen towel (not terry cloth).
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9
Let rolls rise in a draft-free place at warm room temperature until doubled (they will fill pan), 1 to 1 1/2 hours.
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10
Preheat oven to 375u00b0F with rack in middle.
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11
Melt remaining 1 1/2 tablespoons butter.
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12
Brush top of rolls with melted butter and bake until golden-brown, 25 to 30 minutes. Loosen edges with a sharp knife, then transfer rolls to a rack and cool slightly.
