Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans and line the bottoms with parchment paper.

Brown the butter: Place 6 tablespoons butter in a medium saucepan over medium heat. Stir as it melts and foams. Watch closely; when the milk solids turn golden brown and the aroma becomes nutty, remove from heat and cool slightly.

In a large bowl, combine the cooled browned butter with granulated sugar and brown sugar. Beat until combined.

Add eggs one at a time, mixing after each addition. Stir in the vanilla extract.

In another bowl, whisk together flour, baking soda, cinnamon, nutmeg, and salt.

Gradually add the dry mixture to the wet ingredients. Fold in grated carrots and chopped nuts until just combined. Do not overmix.

Divide the batter evenly between the prepared pans. Bake for 30 to 35 minutes or until a toothpick inserted in the center comes out clean.

Let the cakes cool in the pans for 10 to 15 minutes, then invert onto cooling racks and cool completely.

For the frosting: Beat the cream cheese and 4 tablespoons softened butter until smooth. Gradually add the sifted powdered sugar and vanilla, mixing until creamy. Stir in lemon juice if using.

Assemble by placing one cake layer on a serving plate, spread a layer of frosting, top with the second cake, and frost the top and sides. Garnish with chopped nuts or a sprinkle of cinnamon if desired.