In the bowl of a stand mixer, beat together the butter, sugar, and vanilla. When smooth, add the egg, and beat until the consistency's smooth.
Use a spatula to stir in the carrot and sour cream until the mixture's smooth and evenly incorporated.
In a mixing bowl, whisk together the flour, baking powder, spice, and salt.
At a low speed, gradually mix the dry ingredients into the wet until evenly incorporated and the dough's formed.
Wrap the dough in plastic wrap and refrigerate for at least thirty minutes.
Transfer the chilled dough to a lightly floured surface. Knead the dough fully about three times. Roll the dough out evenly until it's about a half inch thick.
Use a 3" donut cutter, cut the donuts out. Reserve the donut holes to cook with the donuts. Re-roll out the scraps as necessary until there's no more.
Fill a dutch oven half way with oil. Heat over medium heat until a deep-frying thermometer reaches 350 degrees. Fry the donuts in batches, about 1 1/2 minutes per side, until golden brown. Remove the cooked donuts to a paper towel lined plate to drain any excess grease.
Fry the donut holes about a minute, or until golden brown.