Preheat your oven to 350°F (175°C) and prepare two 8-inch round cake pans by greasing them with butter or lining them with parchment paper.

In a medium-sized skillet, melt 10 tablespoons of unsalted butter over medium heat. Stir continuously until the butter turns golden brown, around 5-7 minutes.

In a large bowl, whisk together all-purpose flour, baking soda, fine sea salt, ground cinnamon, ground nutmeg, and ground ginger.

In a mixing bowl, combine the cooled brown butter and light brown sugar. Whisk until smooth, then add eggs, unsweetened applesauce, and vanilla.

Gently add the dry ingredients into your wet mixture, stirring until just combined.

Fold in finely shredded carrots and finely chopped walnuts or pecans, if using.

Pour the batter into the prepared cake pans and bake for about 30 minutes, or until a toothpick inserted into the center comes out clean.

Let the cakes cool in the pans for about 10 minutes, then transfer them to wire racks to cool completely.

Prepare the frosting by beating together cream cheese, unsalted butter, vanilla, and powdered sugar until smooth.

Once the cake layers have cooled, frost the layers and serve.