Step 1

Sift 5 cups of the all-purpose flour with 7 teaspoons baking powder into a large bowl and set aside. Whisk 3 large eggs in a small bowl until well-beaten and set aside. Microwave 1 1/2 tablespoons unsalted butter in a small bowl until melted, about 30 seconds.

Step 2

Place 2 cups granulated sugar, 1 1/2 cups slightly warm mashed potatoes, the eggs, and the melted butter in the bowl of a stand mixer fitted with a paddle attachment; beat at medium speed until smooth and well-combined. Add 1 cup whole milk, 1 teaspoon kosher salt, 1 teaspoon nutmeg and mix to combine. Add the flour mixture and mix until incorporated. Cover the bowl with plastic wrap and refrigerate overnight.

Step 3

Make the pumpkin spice coating: Place 1 cup granulated sugar with 2 tablespoons pumpkin spice in a small bowl; whisk until combined and set aside.

Step 4

Prepare the powdered sugar: Place 2 cups powdered sugar in a small bowl and have a small fine mesh sieve on hand. Set aside.

Step 5

Make the vanilla glaze: Place 1 cup powdered sugar, 2 tablespoons milk, and 1 teaspoon vanilla extract in a small bowl. Whisk to combine and set aside.

Step 6

Place 1 1/2 quarts vegetable oil in a 5-quart Dutch oven, attach a candy or deep-fry thermometer, and heat over medium-high heat until the oil is 375 F, 15 to 20 minutes.

Step 7

Sprinkle 1/3 cup all-purpose onto a clean work surface. Place about 1/3 of the dough onto the floured surface; keep remaining dough refrigerated. Using lightly floured hands, knead the dough a couple of times to some of the flour until the dough is stiff enough to handle. Pat dough to about 1/2-inch thickness. Dip a 3-inch doughnut cutter in flour then stamp out doughnuts, dipping the doughnut cutter in flour after each use. If the dough is too soft to cut out, work in some more of the flour from the surface.

Step 8

Use a thin metal spatula coated in flour to carefully transfer 3 to 4 of the formed doughnuts to the hot oil. Fry the doughnuts, turning once, until golden and cooked through, about 2 minutes per side. Transfer to a paper towel-lined baking sheet to drain briefly, then roll or dip the doughnuts in desired coating and set on a wire rack to cool. Working with 8 to 9 doughnut holes at a time, fry the doughnut holes until golden and cooked through, about 90 seconds per side. Transfer to a paper towel-lined baking sheet to drain briefly, then roll or dip in desired coating; set on a wire rack to cool.

Step 9

Repeat shaping, frying, and rolling with the remaining dough. Serve immediately.