Brown the Butter by melting unsalted butter in a medium skillet over medium heat, stirring frequently until golden brown and nutty (5–7 minutes). Cool for 10 minutes.

Mix wet ingredients: Combine cooled brown butter with light brown sugar, eggs, and applesauce in a large bowl, whisking until smooth. Fold in shredded carrots.

Combine dry ingredients: Whisk flour, baking soda, salt, cinnamon, nutmeg, and ginger in a separate bowl until integrated.

Incorporate dry into wet: Gently fold the dry mixture into the wet mixture until just combined.

Bake the Cake: Preheat oven to 350°F (175°C), divide batter between two 9-inch pans, and bake for 30 minutes.

Cool and Slice: Let cakes cool in pans for 10 minutes, then turn onto wire racks to cool completely. Slice each layer in half horizontally.

Frost the Cake: Beat cream cheese and butter until smooth, gradually mix in powdered sugar, and frost each layer.

Garnish: Toast chopped nuts in a skillet and sprinkle on top of the frosted cake.