Preheat the oven to 350 degrees F. Grease two 8-inch cake pans and line the bottoms with rounds of parchment paper.
In a large bowl, mix the flour, brown sugar, cinnamon, nutmeg, baking powder, baking soda, and salt.
In a medium bowl, whisk together the melted butter, eggs, and vanilla.
Make a well in the center of the flour mixture and add the wet mixture. Mix until no dry streaks of flour remain (do not over mix)
Fold in the shredded carrots.
Divide the batter evenly between the prepared pans and bake until a wooden pick inserted into the center comes out clean, 45 to 55 minutes.
Let cool in the pans for 15 minutes, then turn out onto a wire rack to cool completely.
Level off the tops with a cake leveler or a large serrated knife.