Heat oven to 250 degrees. Swaddle coconut in a dish towel to make it easier to grip. Test each of the 3 "eyes" at the stem end to find the softest one. Use a small screwdriver to pierce that "eye," and then completely drain the liquid into a cup. Repeat with remaining coconuts.
Place coconuts on a baking sheet. Heat in the oven for 20 minutes to loosen meat from shells. Wrap a coconut again with the dish towel. Use a hammer to tap the coconut until it cracks, and continue tapping around the broken edge until it begins to break apart. Using a table knife or a screwdriver, carefully pry the white meat from the shell. Using a vegetable peeler, peel away all the brown skin from the white coconut meat. Repeat with remaining coconuts.
Using a Japanese mandoline, thinly slice the coconut meat into thin strips. Transfer to a parchment-lined baking sheet, and toast in the oven just until dry but not browned, about 20 minutes. Set aside to cool.
Cut out one each 14-, 10-, and 6-inch diameter rounds from the foam board to support each tier. Set aside.
Place the bowl of an electric mixer and the whisk attachment in the refrigerator to chill, about 10 minutes. Pour heavy cream into chilled bowl, and return to mixer stand. Whip cream until soft peaks form, about 3 minutes. You may need to work in batches for enough whipped cream to fill the entire cake.
Set one 15-inch-diameter meringue disk on the slightly smaller foam-board round. Spread 1 quart whipped cream on the meringue disk, and sprinkle with 2 cups coconut. Top with the second 15- inch meringue layer, more cream, and more coconut. Add the remaining 15- inch meringue layer, and sprinkle with 2 cups coconut (do not use cream on this layer). Carefully insert 4 dowels through the tier, and set tier aside. Assemble the 11-inch tier in the same manner, using 1 1/2 cups whipped cream and 1 cup coconut per layer, and 1 cup coconut on top layer. Set the 11-inch tier atop the reserved 15-inch tier. Assemble the 7- inch tier in the same manner, using 1 cup cream and 1/3 cup coconut for each layer and 1/3 cup coconut on top layer. Set the 7-inch tier atop the 11-inch tier. Garnish the assembled cake with more coconut and fresh flowers, if desired, and serve.
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