Preheat the oven to 350°F (175°C) and grease the cake pans.

In a saucepan, melt the butter over medium heat until it turns golden brown and has a nutty aroma. Let it cool slightly.

In a mixing bowl, whisk together the granulated sugar, brown sugar, and eggs until well combined.

Add the browned butter to the sugar mixture and mix well.

In another bowl, combine the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.

Gradually add the dry ingredients to the wet ingredients, mixing until just combined.

Fold in the grated carrots and walnuts (if using).

Divide the batter evenly between the prepared cake pans.

Bake for 25-30 minutes, or until a toothpick inserted in the center comes out clean.

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

In a mixing bowl, beat the cream cheese and softened butter until smooth and creamy.

Gradually add the powdered sugar and vanilla extract, mixing until well combined and fluffy.

Once the cakes are completely cool, spread a layer of frosting on top of one cake layer.

Place the second layer on top and frost the top and sides of the cake.

Decorate with additional walnuts if desired.