Preheat the oven to 350°F. Line a standard muffin tin with paper liners.
In a medium bowl, mix together all-purpose flour, whole wheat flour, baking soda, salt, cinnamon, ginger, and nutmeg.
In a large bowl, whisk together the oil, brown sugar, eggs, and vanilla. Stir in grated carrots. Add dry ingredients and mix until just combined.
Divide the batter evenly among the muffin cups (each should be about 2/3 full). Bake until a toothpick inserted into cupcakes comes out clean, 18-20 minutes. Let cool completely.
Make the frosting:
Fit the bowl of a stand mixer with paddle attachment, and beat the cream cheese, butter, and vanilla on medium-high speed until very smooth and creamy. Turn mixer off and scrape down sides of bowl with a rubber spatula. Beat mixture for 2 more minutes on medium-high.
On low speed, gradually mix in the powdered sugar and beat until fluffy. (May need to turn mixer off, scrape down sides of bowl, and continue mixing).
Frost cupcakes using a pastry bag or preferred kitchen tool. Store in an airtight container in the refrigerator for up to 3 days. Store in freezer for up to 2 months.
*Cream cheese and butter need to be at room temperature in order to create a smooth frosting without any lumpiness.