Step 1

Combine the sugar and water in a small saucepan and heat over medium high heat until all the sugar is dissolved. Remove the syrup from the heat. Add the rum and coconut milk and cool to room temperature. The syrup can be made several days ahead and refrigerated

Step 2

Combine the flour, sugar and salt in a small saucepan, whisk to combine. Slowly whisk in the coconut milk until smooth. Heat the mixture over medium low, whisking constantly, until it begins to boil.

Step 3

Continue to cook and stir for 2-3 minutes until it thickens to a pudding-like texture. Immediately remove from the heat. Transfer to a small bowl, stir in the vanilla and the rum, cool to room temperature.

Step 4

Cream the butter until light and fluffy. Add the pudding, a tablespoon at a time, until completely incorporated. Switch to the whisk attachment and whip the buttercream until it’s light and fluffy.

Step 5

Trim the browned edges and slice off the domed top of each cake layer. Split each cake in half horizontally so you have 4 layers.

Step 6

Set aside one half of the frosting for icing the cake, use the rest of the frosting for filling the cake.

Step 7

Place the first cake layer on the serving platter and brush generously with the coconut-rum syrup. Scoop 1/3 of the filling onto the layer and spread evenly. Repeat with the syrup and filling for the next 2 layers. Place the last layer on the cake and brush with syrup.

Step 8

Use the reserved frosting to ice the cake with a thin "crumb coat", then refrigerate until firm. Use the remaining icing to frost the top and sides of the cake. Coat the entire top and sides of the cake with the coconut flakes.

Step 9

Serve at room temperature. Leftovers should be refrigerated.