Preheat oven to 350 degrees. Grease two 6-slot doughnut pans with oil or butter.
In a bowl, beat the sugar and melted butter until combined. Whip in the eggs, one at a time, then the vanilla, Greek yogurt, and carrots. Add the flour, cinnamon, ginger, cloves, baking powder, baking soda, and salt. With the mixer running, stream in the buttermilk. If the batter seems too thick, add milk 1 tablespoon at a time to reach the desired consistency.
Transfer the batter to a piping bag or a ziptop bag with a corner snipped off. Pipe into the prepared donut pans, filling 2/3 of the way full. Bake 10-12 minutes until the centers are set and an inserted toothpick emerges with just a few moist crumbs.
Cool 10 minutes in the pan, then run a knife around the edges of each doughnut and move to a wire rack. Cool completely.
When the doughnuts are cool, make the glaze. Beat the cream cheese, powdered sugar, maple syrup, vanilla, and milk. Add milk as needed, in 1 tablespoon increments, to thin. The glaze should be thick, but not as thick as frosting.
If desired, set aside about 1/3 cup of the glaze and mix with yellow and red food dyes until an orange hue is created.
Lay a piece of parchment paper under a wire rack. Dunk one side of each doughnut into the glaze, letting excess drip back into the bowl. Let the glaze set for 5 minutes. Transfer the orange glaze to a piping bag fitted with a small round tip. Pipe a small dot onto the doughnut, then quickly lift the icing away to create the carrot shape. Repeat as many times as desired. Enjoy!