Preheat oven to 350. Lightly spray donut pan with non-stick cooking spray. If you don’t have a donut hole pan, use a mini-muffin pan instead.
In a medium bowl, stir together flour, brown sugar, white sugar, baking powder, baking soda, salt, cinnamon, nutmeg, and allspice until combined.
In a large mixing bowl beat together butter, eggs, milk, and vanilla extract. Stir in dry ingredients and mix until well combined. Stir in shredded carrots until well incorporated.
Pour batter into a pastry bag or a Ziploc bag with a corner cut off.
Fill donut hole pan or mini cupcake pan just a little over 1/2 full.
Bake for 8-10 minutes.
Cool donuts in pan for 2-3 minutes, then remove to wire rack to let completely cool.