These fluffy, satisfying doughnuts add mashed potatoes to the traditional flour dough -- and are best eaten slightly warm.
Have ready two parchment-lined baking pans. Place potatoes in a small saucepan, cover with cold water, and bring to a boil over high heat. Cook until tender, about 15 minutes. Drain in a colander. Pass potatoes through a potato ricer or food mill and into a medium bowl; set aside.
Place 1/4 cup warm water in a small bowl; sprinkle with yeast, stir gently, and let stand until creamy, 5 to 10 minutes. Using an electric mixer fitted with the paddle attachment, combine the milk, shortening, 1/4 cup sugar, and salt. Let stand until cooled to just warm. Add yeast mixture, reserved potatoes, and eggs; beat until combined. Fit mixer with the dough-hook attachment. Add 5 cups flour, and mix on medium-low speed until combined, adding more flour if necessary, until a smooth and elastic dough is formed, about 5 minutes. Transfer dough to a large, lightly greased bowl; cover. Let stand in a warm place until dough is doubled in size, about 55 minutes.
Transfer dough to a lightly floured surface. Roll out to 1/2 inch thick. Using a 2 1/2-inch-diameter doughnut cutter, cut and place on prepared baking pans. Loosely cover with plastic wrap; let stand in a warm place until dough has risen by about one-third, about 30 minutes.
Combine confectioners' sugar, vanilla, and remaining 6 tablespoons water in a medium bowl, stirring until smooth; set aside. In a medium bowl, whisk together remaining 1/2 cup granulated sugar and cinnamon until well combined; set aside.
In a large, low-sided saucepan over medium heat, heat oil until a deep-frying thermometer registers 375 degrees. Drop doughnuts into oil; fry in batches until golden, 1 to 2 minutes per side. Transfer to several layers of paper towels to drain. Place drained doughnuts on a wire rack set over a baking pan. Dip half the doughnuts in glaze; return to wire rack. Roll remaining doughnuts in the cinnamon-sugar mixture until well coated.
If you do not have a doughnut cutter, use a two-and-a-half-inch cookie cutter to cut out the doughnuts and a three-quarter-inch cookie cutter to cut out the centers.