Line 3 (9-inch) round cakepans with wax paper; lightly grease and flour wax paper. Set pans aside.
Stir together first 4 ingredients.
Beat eggs and next 4 ingredients at medium speed with an electric mixer until smooth. Add flour mixture, beating at low speed until blended. Fold in carrot and next 3 ingredients. Pour batter into prepared cakepans.
Bake at 350°F for 25 to 30 minutes or until a wooden pick inserted in center comes out clean. Drizzle Buttermilk Glaze evenly over layers; cool in pans on wire racks 15 minutes. Remove from pans, and cool completely on wire racks. Spread Cream Cheese Frosting between layers and on top and sides of cake.
Buttermilk Glaze
Bring first 5 ingredients to a boil in a heavy pot or a Dutch oven over medium-high heat. Boil, stirring often, 4 minutes. Remove from heat, and stir in vanilla.
Cream Cheese Frosting
Beat butter and cream cheese at medium speed with an electric mixer until creamy. Add powdered sugar and vanilla; beat until smooth.
Notes
My modifications: I don't buy sweetened coconut. I just use Bob's Red Mill (unsweetened) Shredded Coconut since it really doesn't need to be sweeter. I told you that I didn't line the sides of the pans, but I did use Bak-Kleene non-stick spray, from Williams-Sonoma I find that one batch of the cream cheese frosting is not quite enough, so I make a recipe and a half by using 2 8-oz packages of cream cheese, 4 cups of powdered sugar, 1 cup butter and 2 teaspoons vanilla extract.