Combine 1 cup warm water (105 degrees F to 110 degrees F) and the yeast in a medium bowl; let sit until foamy, about 5 minutes. Sift 1 1/2 cups flour over the yeast; stir until mostly smooth. Cover with plastic wrap and set aside in a warm place until doubled in size with large bubbles on the surface, 20 to 30 minutes.
Meanwhile, prick the potato all over with a fork; microwave until tender, 8 to 10 minutes. Let cool, then peel the potato and grate on the large holes of a box grater. Combine 1 stick butter, the sugar, salt and mustard powder in a stand mixer; beat with the paddle attachment on medium-high speed until light and fluffy, about 2 minutes. Beat in the eggs and grated potato until combined, then beat in the yeast mixture and 1 1/2 cups cheddar until incorporated. Reduce the mixer speed to low and gradually beat in the remaining 2 1/2 cups flour until the dough starts coming together in a soft, sticky ball. (You may not need all of the flour.) Gather into a loose ball and transfer to a lightly oiled bowl; turn to coat. Cover with plastic wrap and set aside in a warm place until doubled in size, about 1 1/2 hours.
Preheat the oven to 350 degrees F. Butter a 9-by-13-inch baking dish. Turn the dough out onto a lightly floured surface and divide into 24 pieces. Shape each piece into a ball, dusting with flour as needed, and arrange in the baking dish, slightly touching. Set aside until the rolls are lightly puffed, about 20 minutes.
Lightly oil a sheet of foil and loosely cover the rolls, oiled-side down. Bake until just firm to the touch, about 30 minutes. Uncover, brush with the remaining 2 tablespoons butter and sprinkle with the remaining 1/2 cup cheddar. Continue baking until the rolls are golden brown and the cheese is melted, about 20 more minutes. Let cool at least 20 minutes before serving.
Photograph by Con Poulos
Recipe courtesy of Food Network Magazine