Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13 inch pan.
In a medium bowl, mix together flour, baking soda, cinnamon, salt and nutmeg.
In a large bowl, beat together eggs, oil, white sugar and 2 teaspoons vanilla extract until light and creamy.
Stir in the flour mixture and carrots until just blended.
Pour into the prepared pan.
Bake for 35 to 40 minutes in the preheated oven. Cake is done when it springs back to the touch.
Allow to cool in pan for 10 minutes, then turn out onto a wire rack and cool completely.
In a medium bowl, combine cream cheese, butter or margarine, remaining 1 teaspoon vanilla extract and confectioners' sugar. Beat until smooth. Spread over cooled cake.