Brown the butter, then transfer to the fridge to chill for 45–60 minutes until firm.
Preheat oven to 350°F. Line a 12-cup cupcake pan with paper liners. Set aside.
In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, cloves, nutmeg, ginger, and salt until combined.
In a separate large bowl, whisk together oil, granulated sugar, and brown sugar.
Add eggs, sour cream, and vanilla. Whisk until smooth.
Using a spatula, gently fold the dry ingredients into the wet ingredients. When mostly combined, add in the grated carrots and fold until no dry pockets remain. Do not overmix.
Scoop batter into cupcake liners, filling each about ⅔ full.
Bake for 16–17 minutes, until a toothpick inserted in the center comes out clean. Let cupcakes cool completely before frosting.
For the frosting, beat the chilled brown butter and cream cheese on medium-high speed until light and fluffy (about 2 minutes).
Reduce speed to low and add powdered sugar, 1 cup at a time, mixing after each addition until fully incorporated.
Add vanilla and salt, then increase speed to high and beat for 1 minute.
Transfer frosting to a piping bag and pipe onto cooled cupcakes, or spread with a knife/offset spatula. Decorate as desired.
