Potato Doughnuts

Name: Potato Doughnuts
Ingredients: 6 cups all-purpose flour, divided; 7 teaspoons baking powder; 3 large eggs; 1 1/2 tablespoons unsalted butter; 2 cups granulated sugar; 1 1/2 cups slightly warm plain mashed potatoes (see Recipe Notes); 1 cup whole milk; 1 teaspoon kosher salt; 1 teaspoon ground nutmeg; 1 1/2 quarts (48 ounces) vegetable oil, for deep frying; 1 cup granulated sugar; 2 tablespoons pumpkin pie spice; 2 cups powdered sugar; 1 cup powdered sugar; 2 tablespoons whole milk; 1 teaspoon vanilla extract