Preheat an oven to 350 F. Grease and line two 9-inch round cake pans with parchment paper.
Brown the butter. Add the butter to a small saucepan over medium-low heat. Let the butter melt and cook, swirling the pan every so often, until the butter is golden brown and smells nutty. Transfer to a heat-safe large glass mixing bowl and let it cool slightly.
Mix the dry ingredients. In a separate mixing bowl, whisk together the flour, baking soda, kosher salt, milk powder, cinnamon, nutmeg, ground ginger, and cloves. Set aside.
Mix the wet ingredients. Add the oil, brown sugar, molasses, and granulated sugar to the bowl with the slightly cooled brown butter. Whisk for 2 minutes, until the oil is well incorporated into the sugar. Whisk in the eggs, sour cream, and vanilla bean paste until very smooth. Switch to a rubber spatula and fold in the flour mixture, grated carrots, and walnuts until the batter has no streaks of flour.
Bake the cakes. Evenly distribute the batter between the prepared cake pans. Bake for 30-35 minutes, until a toothpick inserted comes out clean. Let the cakes cool completely before removing them from the cake pans. If the cakes are domed, you can level off the tops with a serrated knife.
Make the frosting. In a mixing bowl, combine the softened butter, vanilla, and cream cheese and use a hand mixer to beat them together for 2 minutes, until smooth. Gradually beat in the powdered sugar, adding cream as necessary when the mixture gets too thick. Whip the frosting until it’s smooth and spreadable.
Frost the cake. Spread an even layer of frosting between the two cake layers. Then, apply a thin layer of frosting to the outside of the cake. Refrigerate for 20 minutes. Then, spread a thick and even layer of frosting around the rest of the cake. You can reserve some frosting to pipe decoratively around the cake. Let the cake chill for 1 hour before serving.