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

Brown the butter: place butter in a medium saucepan over medium heat. Stir constantly as it foams, then darken to golden and smell nutty. Remove from heat and cool slightly.

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

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

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

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

Divide the batter evenly between the prepared pans. Smooth the tops.

Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.

Cool the cakes in the pans for 10 minutes. Run a knife around the edges, invert onto a cooling rack, and remove the parchment. Cool completely before frosting.

Make the frosting: beat cream cheese and softened butter until smooth. Gradually add powdered sugar, then mix in vanilla extract and lemon juice if using. Beat until creamy and spreadable.

Place one cooled layer on a cake plate. Frost the top. Add the second layer and frost the top and sides as desired. Garnish with chopped nuts or a light dusting of cinnamon if you like.