In a medium-sized bowl, whisk together the flour, baking powder, cinnamon, nutmeg, salt, and cloves. Set aside.
In a large bowl, whisk together the melted butter, egg, brown sugar, milk, yogurt, and vanilla extract until combined.
Add the dry ingredients into the wet and gently fold together using a silicone spatula or wooden spoon until combined (be careful not to over mix). Then gently fold in the grated carrots.
Scoop the batter into a piping bag or a zipped-up sandwich bag. Simply cut a corner off of the bottom of the bag and pipe the donut batter into each donut cup, filling it up about 3/4 of the way full.
Bake the donuts for 9-11 minutes (10 minutes is perfect for me). To check if the donuts are done, touch the top of one carefully with your finger. If it bounces back, they're done. You could also take a toothpick to check the center. If it comes out clean, then they're done. Allow the donuts to cool in the pan for 5 minutes before carefully removing them and setting them on a wire rack to finish cooling.