Preheat oven to 350 degrees. Butter and flour a 5-by-9-inch (6-cup) loaf pan. In a bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Using an electric mixer, beat butter and brown sugar until light and fluffy; beat in eggs and 1 teaspoon vanilla. Beat in carrots. With mixer on low, gradually add flour mixture; beat just until combined. Transfer batter to prepared pan; bake until a toothpick inserted in center comes out clean, 40 to 45 minutes. Let cool in pan 5 minutes. Turn cake out onto a wire rack, and let cool completely.
Make frosting: Using mixer, beat cream cheese, confectioners' sugar, and 1/2 teaspoon vanilla until fluffy. Frost top of cooled cake.