Prepare the Potatoes: Peel, dice, and boil the potatoes in a saucepan until they are fork-tender. Drain, mash thoroughly, and measure out 1 cup.
Mix with Butter: In a large mixing bowl, combine the mashed potatoes and melted butter, stirring until well blended.
Incorporate Eggs and Sugar: In a separate bowl, whisk together the eggs and sugar, then add this mixture to the potato mixture, beating until smooth.
Activate the Yeast: Stir in the dissolved yeast mixture.
Prepare Dry and Wet Ingredients: In another bowl, whisk together the flour, nutmeg, salt, baking powder, and baking soda. In a separate container, mix the buttermilk and vanilla extract.
Combine Everything: Gradually add the dry ingredients and the buttermilk mixture to the potato mixture, alternating between the two, stirring until a soft dough forms.
Let It Rest: Cover the bowl with a clean kitchen towel and let the dough rise for about an hour.
Roll and Cut: Flour a work surface and roll out the dough to about ½-inch thick. Use a floured doughnut cutter to cut out shapes.
Fry to Perfection: In a heavy-bottomed pot or Dutch oven, heat oil to 375°F. Fry the doughnuts in small batches, cooking for about 2 minutes per side until golden brown. Remove and drain on a wire rack lined with paper towels.
Coat with Cinnamon Sugar: While still warm, roll each doughnut in the cinnamon-sugar mixture to coat.