In a small saucepan, cover carrots with water. Bring water to a boil and cook until tender, about 15 minutes. Let cool and mash.
In a large mixing bowl, beat eggs. Add vegetable oil, buttermilk, 2 cups white sugar and 2 teaspoons vanilla. Add flour, baking soda, ground cinnamon and salt. Mix in pineapple, 2 cups mashed carrots, nuts and coconut.
Pour into paper towel-lined 13 x 9 x 2-1/2 inch pan.
Bake at 350 degrees F (175 degrees C) for 55 minutes, or until cake tests done.
Place cake on a plate, and allow to cool completely before frosting.
To Make Frosting: Combine cream cheese, melted butter, 1 teaspoon vanilla extract, and confectioners' sugar. Spread on cooled cake. Cake tastes best after refrigerating overnight.