Brown the butter: Place unsalted butter in a medium saucepan over medium heat. Stir frequently until foamy and milk solids turn golden brown (about 5-7 minutes). Remove from heat and cool slightly.

Mix wet ingredients: In a large bowl, combine browned butter, granulated sugar, and brown sugar. Beat until well mixed. Add eggs one at a time. Stir in vanilla extract.

Combine dry ingredients: In a separate bowl, whisk together flour, baking soda, cinnamon, nutmeg, and salt. Sift if desired.

Incorporate carrots and nuts: Gradually add dry ingredients to the wet mixture, folding gently until just combined. Fold in grated carrots and nuts if using.

Preheat oven to 350°F (175°C). Grease and flour two 9-inch round cake pans. Divide batter evenly and smooth tops.

Bake for 30-35 minutes or until a toothpick comes out clean and the cake springs back when touched. Rotate pans halfway through baking for even color.

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

Make frosting: Beat cream cheese and butter until smooth. Gradually add sifted powdered sugar and vanilla extract. Optionally, add lemon juice.

Assemble cake: Place one layer on a serving plate, frost the top, then add the second layer and frost the top and sides. Optionally sprinkle with nuts or cinnamon.