A traditional fallback in times of grain shortage, potato doughs yield rolls that have a wonderfully crisp crust and a complex, deep flavor. Yeast cells are very fond of potato starch, and produce moist, airy rolls.
Place potatoes in a medium saucepan; cover with cold water. Bring water to a boil; reduce to a simmer. Cook potatoes until knife-tender, about 15 minutes. Drain, reserving 1/2 cup of the liquid. Mash potatoes; set aside.
Cool reserved liquid to 105 degrees. In the detached bowl of an electric mixer, whisk liquid, yeast, and pinch of sugar. Set aside until mixture is foamy, about 10 minutes.
Attach bowl to electric mixer fitted with the dough-hook attachment. With the mixer on low, add remaining sugar, reserved potatoes, buttermilk, 4 tablespoons melted butter, and salt. Gradually add enough flour to make a slightly sticky dough. Knead dough until smooth, about 2 minutes. Brush a mixing bowl with butter. Place dough in bowl; cover bowl with buttered plastic wrap. Set aside to rise until doubled, 1 to 1 1/2 hours.
Heat oven to 375 degrees. Line 3 large baking sheets with parchment. Turn dough out onto a floured work surface. Using a floured rolling pin, roll dough out 3/4 inch thick. Cut dough into 2-inch-wide strips. Cut strips into triangles or squares; place at least 1 1/4 inches apart on baking sheets. Brush tops with remaining melted butter; cover with buttered plastic wrap. Let rise until dough does not spring back when pressed with a finger, 15 minutes. Bake until golden, 18 to 20 minutes; serve.