In a medium bowl, combine the water, yeast, and 1 1/2 cups (6.75 ounces) of the flour and stir until combined. 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 the butter, 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. Shape into a loose ball and transfer to a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise until doubled in size, about 1 1/2 hours.
Preheat the oven to 350°F. Spray a 9-by-13-inch baking dish with nonstick cooking spray. 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 so they are slightly touching. MAKE AHEAD: At this point the shaped rolls can be covered and refrigerated overnight, let sit at room temperature for 1 hour or until puffy. If already at room temperature, let rise for about 30 minutes, or until puffy.
Spray a sheet of foil with nonstick cooking spray and loosely cover the rolls, oiled-side down. Bake until just firm to the touch, about 30 minutes. Uncover, brush with the 2 tablespoons melted butter and sprinkle with the remaining 1/2 cup cheddar. Continue baking, uncovered, until the rolls are golden brown and the cheese is melted, about 20 more minutes. Let cool at least 20 minutes before serving.
Another make-ahead option: Parbake, covered, for 30 minutes, then let cool and refrigerate overnight. Bring to room temperature, brush with the butter and bake, uncovered, 40 minutes, sprinkling with 1/2 cup cheddar during the last 15 minutes.