Preheat oven to 350ºF. Grease a mini bundt pan with nonstick spray and set aside.

In a medium-sized bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, allspice, and salt. Set aside.

In a large bowl, whisk together the melted butter, applesauce, eggs, vanilla extract, brown sugar, and granulated sugar until combined. Add the dry ingredients and whisk until combined. Fold in the grated carrots using a spatula. 

Pour the batter into the prepared pan, filling each cavity about 2/3 fill. Bake at 350ºF for 15-17 minutes, or until a toothpick inserted into the center comes out clean. Allow the cakes to cool in the pan for about 10 minutes before transferring to a wire rack. Allow the cakes to cool completely before frosting and allow the pan to cool completely before greasing again and baking any remaining cakes.

Heat the butter in a skillet (preferably one with a light colored bottom so it is easy to tell when the butter is done browning) over low heat until browned. Allow to cool for 5 minutes.

In a medium-sized bowl, use an electric handheld or stand mixer to beat the cream cheese on medium speed for about 1 minute, until smooth. Add the cooled brown butter, powdered sugar, and vanilla and beat on low speed until combined. To frost the mini bundt cakes, cut a small corner off of a ziplock bag and pour the frosting in. Once the cakes have cooled completely, you can frost them to your liking.