Prepare the Cheesecake Layer: Arrange oven racks to lower and middle positions and preheat oven to 325ºF. Place a 9x13-inch baking dish or pan on the lower rack of oven and fill it halfway with water. The middle rack should be just above the pan of water which will provide added moisture to help prevent the cheesecake from cracking during baking. Generously grease springform pan with butter and set aside.
Using an electric stand mixer fitted with paddle attachment, or handheld mixer and large bowl, beat the cream cheese until creamy. Gradually add sugar and beat until fully incorporated, scraping down the sides of bowl as necessary with rubber spatula. Add the eggs, one at a time, blending well after each addition. Add the sour cream, heavy cream and vanilla; mix until fully incorporated and smooth. Using rubber spatula, scrape and pour batter into prepared pan. Place springform pan on middle rack in preheated oven above the pan of water.
Bake in preheated oven until cheesecake is just set in center and no longer jiggles, about 40 to 45 minutes. Remove cheesecake from oven and place on wire cooling rack. Immediately, run thin metal icing spatula around edge of pan to loosen cheesecake from sides of pan. This will prevent any surface cracking as the cheesecake cools and contracts. Allow cheesecake to cool completely, at least 1 hour. Once the cheesecake has cooled, place it (while still in the springform pan) into the freezer, covered with plastic wrap. Let the cheesecake freeze completely, about 4 to 5 hours or overnight.
Prepare the Carrot Cake Layers: Arrange oven rack in center of oven and preheat oven to 325º F. Grease two 9-inch round cake pans with butter or oil. Line bottoms with parchment or wax paper circles. Grease paper linings and dust lined pans with flour, tapping out the excess.
Using an electric stand mixer fitted with paddle attachment, or handheld mixer and large bowl, mix flour, sugars, baking powder, baking soda, salt and spices on low speed (stir) until well combined. Add oil, eggs and vanilla; mix on low speed (stir), scraping down the sides of bowl as necessary with rubber spatula. Beat on medium speed until well blended, about 2 minutes. Fold in carrots, coconut and pineapple and mix until thoroughly combined. Using rubber spatula, scrape and pour batter into prepared pans, dividing evenly (about 3 cups of batter per cake pan). Smooth tops evenly with rubber spatula.
Bake in center of preheated oven until a toothpick inserted into center comes out clean, about 40 to 45 minutes. Cool cakes in pans on wire racks for 10 minutes. Run thin metal spatula around edges of each cake to loosen, then invert onto wire rack and peel away parchment or wax paper. Turn right side up onto wire racks (using an additional rack to flip); cool completely.
Prepare the Cream Cheese Buttercream: Using an electric stand mixer fitted with paddle attachment, or handheld mixer and large bowl, beat cream cheese, butter and vanilla on medium speed. Gradually increase mixer speed to high and continue beating until light and fluffy, scraping down the sides of bowl as necessary with rubber spatula. Gradually add 4 cups confectioners’ sugar, beating on low speed (stir), until well combined. Add additional confectioners’ sugar until desired consistency for spreading and level of sweetness is reached. Beat on high speed until well combined and smooth, about 1 minute.
Assemble & Ice the Cake: Place one cake layer onto serving plate, platter or cake pedestal or stand in the center. With icing spatula, spread about 3 tablespoons frosting evenly over top of cake. Remove cheesecake layer from the freezer. Loosen cheesecake from sides of pan by running thin metal spatula around the inside rim. Unmold cheesecake by removing sides of pan. Carefully, slide a hot knife (warmed under hot running water and wiped dry) under the cheesecake, and slide knife around the circumference, to release it from bottom of springform pan. (I used my chef’s knife for this task. However, a long serrated knife will work as well. Be careful!)
If necessary, trim cheesecake to match the size of the cake layers. This is done by measuring the cheesecake layer against the cake layers on a cutting board. (I did this by using a sheet of wax paper between the cheesecake layer and the remaining carrot cake layer on cutting board.) If the cheesecake layer is slightly larger in circumference, use a knife to carefully shave away some of the outer edge of the frozen cheesecake to match the outer edges of the cakes until all sides are even or flush. (When trimming the cheesecake layer, I lifted the wax paper upward to take a peek and then sliced the cheesecake using my chef’s knife.)
Place cheesecake layer on top of the first cake layer. Spread 3 tablespoons frosting over top of cheesecake. Place second cake layer on top of the cheesecake layer. Spread frosting over top and sides of cake neatly and evenly. You may do a crumb coat and then chill the cake for 30 to 45 minutes before completing the icing, if desired. However, if decorating the sides of the cake with nuts, you can skip the crumb coat because the nuts will hide any crumbs showing.
To Decorate the Cake: If decorating the cake, gently press finely chopped walnuts onto sides of cake. This is easy to do if the cake layers are assembled onto a cardboard cake round. Simply hold and balance the cake on the cake round in one hand and use free hand to press the nuts into the sides of the cake. Use a sheet of parchment or wax paper, or even a sheet pan, to catch the excess.
If not using a cardboard cake round to hold the cake and press nuts into sides of cake, it is still easy to do. Simply line your cake plate or pedestal with four 4- to 5-inch wide strips of wax paper and use hand to press nuts onto sides of cake. The wax paper strips will catch the nuts that fall away. Turn the cake as you work (a cake turntable is helpful, but not necessary) all the way around. Once finished, scoop away the fallen nuts with spoon and pull the wax paper sheets away leaving a clean cake plate or pedestal.
Pipe a simple or fancy border with your favorite tip (I used a no. 21 star tip) around the perimeter of the top layer. Sprinkle shredded coconut in a circle on top of the cake, just inside the piped border. Then follow with freshly grated carrot just inside the perimeter of the coconut. Dust the top of the cake with freshly grated whole nutmeg.
To Store the Cake: Keep cake covered and stored in refrigerator until ready to slice and serve.
To Slice & Serve the Cake: Slice the chilled cake with a thin knife that has been dipped in hot water. Wipe knife clean after each cut. Use a cake server to transfer slices of cake to dessert plates. Serve immediately.