Preheat your oven to 350°F and prepare a cupcake pan by adding liners and set aside.
In the bowl of a stand mixer equipped with a whisk attachment, add the eggs, vegetable oil, applesauce, both sugars, vanilla and mix until just combined.
Add in the flour, baking powder, salt, cinnamon, and nutmeg and mix until just combined and there are no dry spots remaining.
Fold in the grated carrots until evenly distributed through the batter.
Use a cookie scoop to divide the batter evenly between 15 cupcake liners, filling about ¾ full.
Bake for 20-23 minutes or until a toothpick inserted in the center of the cupcakes come out clean.
Remove from the oven and allow to cool in the pan for 5 minutes then transfer to a wire rack to cool completely while you prepare the frosting.
Frosting:
Add the room temperature butter and cream cheese into a bowl of a stand mixer equipped with a paddle attachment and cream for 2-3 minutes or until smooth.
Add the vanilla and mix until just combined.
Add the powdered sugar one cup at a time and mix until well combined. Slowly increase the speed of the mixer to high and mix on high for 2-3 minutes until the frosting is nice and fluffy.
Add the frosting to a piping bag with a large round tip (I used a Wilton 1A).
Assembly:
Pipe a small circle of frosting on the top of each cupcake and carefully dip the top of each cupcake into crushed walnut pieces. Pipe another generous swirl of frosting on top of the walnuts and sprinkle with grated carrot.
Store in an airtight container in the refrigerator for up to three days.
If you've tried this recipe, come back and let us know how it was in the comments or ratings.