Preheat oven to 350 degree F. Line cupcake tin with paper liners. Set aside.
Mix sugar, oil, vanilla, and eggs. Add flour, baking soda, baking powder, salt, ginger, cloves, nutmeg, and cinnamon. Beat for 2 minutes until fully combined and fluffy.
Fold in carrots and chopped nuts.
Fill cupcake tins 2/3 full. Bake for 20-22 minutes. Remove and cool completely before frosting.
For the frosting, cream together butter and cream cheese (about 3 minutes). Add vanilla and powdered sugar, beat well. Use an open tip to pipe onto cupcakes.
Notes
Carrots. You can grate carrots for carrot cake using either a box grater or a food processor. Make sure to start with washed, peeled carrots. If using a box grater, place the carrot against the side of the grater and run the carrot downward to grate. When you use a food processor, use the grating attachment. You want the pieces very fine without turning into a puree.
Use a stand mixer or beaters. An electric mixer is the easiest (and fastest) way to make fluffy spreadable cream cheese frosting.
Fill cupcake liners 2/3 of the way. Avoid overfilling the cupcake tins. These cupcakes don't dome up on top as much as traditional chocolate cupcakes. The heaviness from the carrots keeps them flatter. Still delicious but much easier to frost.
These can be made with or without nuts. Simply omit the nuts from the batter.
Store. Frosted cupcakes should be stored in airtight container in the refrigerator, for up to 5 days.