Preheat oven to 350 ℉. Grease and line a 9x9 pan with parchment paper. Set aside. To make the browned butter, melt the butter in a small saucepan. Once melted, stir constantly until the butter turns dark brown. Pour into a heat safe bowl to cool.
In a large bowl, whisk together the butter, sugars, vanilla, eggs and Greek yogurt, In a medium bowl, stir together the flour, baking powder, baking soda and spices, Add the flour mixture to the wet mixture along with the buttermilk, carrots and hot water. Mix until combined.
Pour the batter into the prepared pan and bake for about 30 minutes or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for about 10 minutes before transferring to a cooling rack to cool completely.
To make the frosting, beat together the cream cheese, butter and vanilla until light and fluffy. Add the powdered sugar and beat again until smooth. Set aside one or two spoonfuls into a separate bowl and color it orange. Spread the remaining frosting onto the cake (see the post caption above for detailed instructions on how I piped the carrots)
Cut the cake into 9-12 slices and enjoy!