1. Place potatoes in medium saucepan and add water to just cover. Bring to boil over high heat; reduce heat to medium-low and simmer until potatoes are cooked through, 8 to 10 minutes.

2. Transfer 5 tablespoons potato water to bowl to cool; drain potatoes. Return potatoes to saucepan and place over low heat. Cook, shaking pot occasionally, until any surface moisture has evaporated, about 1 minute. Remove from heat. Process potatoes through ricer or food mill, or mash well with potato masher. Measure one very firmly packed cup potatoes (8 ounces) and transfer to bowl. Stir in butter until melted.

3. Combine flour, 1/2 cup Cheddar, yeast, sugar, salt, mustard, and cayenne in bowl of stand mixer. Add warm potato mixture to flour mixture and mix with hands until combined (some large lumps are OK). Add 1 egg and reserved potato water; mix with dough hook on low speed until dough is soft and slightly sticky, 8 to 10 minutes.

4. Shape dough into ball and place in lightly greased container. Cover tightly with plastic wrap and allow to rise at room temperature until almost doubled in volume, 30 to 40 minutes.

5. Turn out dough onto counter, dusting with flour only if dough is too sticky to handle comfortably. Pat gently into 8-inch-square of even thickness. Using bench knife or chefs knife, cut dough into 12 pieces (3 rows by 4 rows). Separate pieces and cover loosely with plastic.

6. Line rimmed baking sheet with parchment paper. Working with 1 piece of dough at a time and keeping remaining pieces covered, form dough pieces into smooth, taut rounds. (To round, set piece of dough on unfloured work surface. Loosely cup hand around dough and, without applying pressure to dough, move hand in small circular motions. Tackiness of dough against work surface and circular motion should work dough into smooth, even ball, but if dough sticks to hands, lightly dust fingers with flour.) Arrange rolls on prepared baking sheet. Cover loosely with plastic and let rise at room temperature until almost doubled in size, 30 to 40 minutes. While rolls rise, adjust oven rack to upper-middle position and heat oven to 425°F.

7. Brush rolls gently with egg wash. Sprinkle each roll with 1 teaspoon Cheddar (1/4 cup total). Bake rolls until deep golden brown, 12 to 14 minutes, rotating baking sheet halfway through baking. Transfer baking sheet to wire rack and cool for 5 minutes. Transfer rolls from baking sheet to wire rack. Serve warm or at room temperature.

Makes 12 Rolls