Preheat the oven to 350 degrees. Prepare two oven-safe donut pans with nonstick cooking spray. In a large mixing bowl, whisk together the dry ingredients: flour, baking powder, cinnamon, ginger, nutmeg, and salt.
In a separate bowl, whisk together the buttermilk, oil, brown sugar, egg, and vanilla extract until well blended.
Pour the wet ingredients into the dry ingredients and stir just until combined. Fold in the carrots.
Transfer the batter to a piping bag or gallon-sized freezer bag with the end snipped. Squeeze the batter into the prepared doughnut pan, dividing the dough between 10 or 12 molds depending on how thick you desired the doughnuts to be. Bake for 10 to 12 minutes or until a toothpick inserted in the middle of the cake returns clean.
Place the doughnut pans on a wire rack to cool for 5 minutes before flipping the doughnuts out to continue cooling.
Meanwhile, beat the cream cheese and butter until smooth. Add the powdered sugar, vanilla extract, and milk. Beat until creamy.
Pipe or spread the frosting over the doughnuts. Sprinkle with chopped pecans if desired. In place of the pecans, you could use sprinkles, or use both!