Combine 1 cup flour, sugar, dry yeast, and salt in a larger mixer bowl. Heat water, milk, and butter until very warm (120° to 130°F). Stir in potato flakes; let soften 1 minute. Stir into dry ingredients. Stir in 1 egg and enough remaining flour to make soft dough. Knead on lightly floured surface until smooth and elastic, about 4 to 6 minutes. Cover; let rest on floured surface 10 minutes.
Divide dough into 12 equal pieces. Roll each piece into a ball. Arrange rolls in a greased 13 x 9-inch baking pan. Place large shallow pan on counter; half fill with boiling water and top with wire rack. Place baking pan on wire rack over hot water, making sure the pan doesn't come in contact with the water. Cover; let rise 15 minutes or until double in size.
Lightly beat remaining egg; brush over rolls. Bake at 375°F for 25 to 27 minutes or until golden brown. Remove to a wire rack to cool.
*To use fresh potato: Follow above directions except decrease water to 1/2 cup and replace potato flakes with 1/3 cup cooked, mashed potato (at room temperature). Stir mashed potato into dough along with egg.