Preheat the oven to 300F. To make the macaroons, beat the egg whites until they have soft peaks, then beat in half the sugar, 2 tablespoons at a time. Fold in the flour, ground almonds, and remaining sugar. Line 2 baking sheets with parchment paper and draw the outline of the following circle sizes on the paper: 4 inches, 5 inches, 6 inches, 7 inches. Place a medium-round smooth nozzle in a pastry bag and fill it with the meringue. Pipe concentric circles of the meringue into the outlined circles, starting in the middle. Bake for 25 minutes. Make a classic pastry cream with the given ingredients and dot it with 2 tablespoons of the butter. Cool. Add the warmed rum and remaining softened butter, beating well to combine. To assemble: spread a layer of the pastry cream on the largest meringue circle. Place the 6-inch meringue circle on top of the pastry cream. Cover with a layer of pastry cream. Repeat layers, finishing with the smallest meringue round. Spread the remaining pastry cream over the top and sides. Sprinkle with the cocunut and sugar almonds. Serve chilled.
