Preheat the oven to 350F. Add the brown sugar, canola oil, greek yogurt, eggs, and vanilla. Whisk until completely combined.
Add the flour, cinnamon, baking powder, baking soda, and salt. Stir until a few streaks of flour remain.
Peel and grate the carrots. Transfer the grated carrots to a paper towel and squeeze out the juice -- this removes extra, unwanted moisture. Add the carrots to the batter and stir until combined.
Line a muffin tin with baking cups. Scoop the batter into the baking cups until they're about 3/4 full. Transfer to the oven and bake for 18-20 minutes, until an inserted toothpick comes out clean.
While the muffins cool, make the frosting. Add the butter and cream cheese to a bowl. Using a stand mixer or hand mixer, mix for 3-4 minutes until the mixture is smooth. Add the powdered sugar and vanilla and mix an additional 2-3 minuets until the frosting is smooth.
Transfer the frosting to a piping bag and pipe a couple tablespoons of frosting on top of each cupcake. Use an offset spatula (or butter knife) to spread the frosting across the top of the cupcake.
Keep any extra cupcakes in the fridge and enjoy!