The brown butter will need time to cool and solidify again before using, so begin with this step.
Melt butter over medium heat in a medium saucepan. Once the butter has melted, let it cook for around 3-4 minutes, stirring regularly until the milk solids have caramelized into a golden amber color and the butter has a nutty aroma.
Pour the melted butter into a large heat-proof bowl, including all the brown bits. Let this cool until it has solidified at room temperature. You can place it in the refrigerator to speed up the solidifying process, but make sure it's at room temperature when you're ready to use it.
Cake
Preheat the oven to 350°F /180°C and grease, and line a 9x13-inch pan with parchment paper.
Combine flour, baking powder, soda, salt, and warm spices in a large bowl.
Mix brown sugar, white sugar, eggs, vegetable oil, yogurt, and vanilla extract in a separate bowl. Mix in the grated carrots and chopped nuts.
Add the flour mixture to the wet ingredients, and fold them with a wooden spoon or rubber spatula.
Pour the batter into the prepared pan. Spread it out evenly, and bake for 32-35 minutes or until a wooden pick inserted in the center of the cake comes out clean. Let the cake cool on a wire rack.
Frosting
When the brown butter has solidified, beat it with an electric hand mixer or stand mixer with a whisk attachment on high speed until light and creamy.
Add room-temperature cream cheese and beat it until fluffy.
Add powdered sugar, vanilla, and salt and beat it on low speed until the sugar is combined, then switch to high speed and beat until creamy. Add a tablespoon of milk or cream if the frosting is too thick. If too thin, add a bit more powdered sugar.
Spread the frosting over the cooled cake.
Notes
Measurements: Both grams and cup sizes are available for this recipe. Just use the toggle on the recipe card.Variations: This cake can also be turned into a 2-layer cake using two 8-inch round cake pans. You can add extra flavor with crushed pineapple or raisins, dried apricots, or dried cranberries.Make Ahead: The cake can be baked, cooled, and covered at room temperature overnight. Frosted cake or unfrosted cake layers can be frozen for up to 3 months.