Preheat the oven to 350℉. Spray a 9x5 (or 8x4) inch pan with nonstick spray.
In a small bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon and nutmeg.
In a large bowl, use an electric mixer to combine the butter and brown sugar until light and fluffy; beat in the eggs and vanilla, then the carrots. Add the dry ingredients and beat just until combined. Scrape the batter into the prepared pan.
Bake 40 to 45 minutes. Let cool in the pan for 15 minutes, then turn the cake onto a rack to cool completely.
To make the frosting, use an electric mixer to combine the cream cheese, powdered sugar and vanilla until creamy and well mixed. Spread the frosting on top of the cooled cake. Keep refrigerated in a covered container if you don't eat it all right away!