Preheat oven to 400℉.
Prick potato all over with a fork and place directly on oven rack. Bake until fork-tender, 40-60 minutes. Let cool slightly, then cut in half, scoop potato flesh from the skin into a bowl, and mash until completely smooth. Measure 1 cup mashed potato and set aside.
In a medium bowl, whisk to combine flour, baking powder, baking soda, salt, and nutmeg.
In a large bowl, beat eggs with a hand mixer on high speed until light and foamy. Beat in sugar until thick and fluffy. On low speed, beat in milk, butter, and vanilla. Beat in mashed potatoes.
Stir flour mixture into potato mixture until smooth.
On a generously-floured countertop, roll out the dough to ½” thick. Use a doughnut cutter to cut out as many doughnuts as possible (or use one 2½-2¾” round cutter and one 1” round cutter to cut the doughnuts and doughnut holes). Re-roll dough scraps 1 time and cut out more doughnuts.
Add oil to a large, heavy pot or Dutch oven to a depth of 3”. Heat oil to 365℉.
While oil heats, in a large bowl whisk to combine sugar, cinnamon, cardamom, nutmeg, and salt. Set aside.
Fry doughnuts and doughnut holes in batches (do not crowd the pot) until golden brown on the bottom, 2-3 minutes, then carefully turn and fry second side. Try to keep the oil temperature as consistent as possible.
Transfer doughnuts to a baking sheet lined with paper towels to drain briefly. While still warm, toss in cardamom sugar.