Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch pan.
Beat together eggs, oil, white sugar, and 2 teaspoons vanilla extract in a large bowl. Mix in flour, baking soda, baking powder, salt, and cinnamon. Stir in carrots. Fold in pecans. Pour into prepared pan.
Bake in preheated oven until a toothpick inserted into center of cake comes out clean, 40 to 50 minutes. Let cool in pan for 10 minutes, then turn out onto a wire rack and cool completely.
To make frosting: Combine butter, Neufchatel cheese, confectioner's sugar, and 1 teaspoon vanilla extract. Beat until mixture is smooth and creamy. Stir in chopped pecans. Spread frosting over completely cooled cake.
