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.