Heat the oven to 325°F and arrange a rack in the middle.
Coat a 13-by-9-inch glass baking dish with butter; set aside.
Place the flour in a small bowl and whisk to aerate and break up any lumps; set aside.
Separate the eggs, placing the yolks in the bowl of a stand mixer.
Reserve the whites in a separate, very clean, medium bowl.
Add the sugar to the yolks and, using the paddle attachment, beat on high speed until pale yellow, about 5 minutes.
Transfer the mixture to a large bowl; set aside.
Thoroughly clean and dry the stand mixer bowl. Place the egg whites in the clean bowl and, using the whisk attachment, whip on high speed until medium peaks form, about 1 1/2 minutes.
Using a rubber spatula, stir about a third of the egg whites into the yolk mixture to lighten it. Then gently fold in the remaining whites.
Sprinkle the flour over the egg mixture and gently fold it in, just until there are no more white flour streaks. (Do not overmix)
Pour the batter into the prepared baking dish and bake until the cake is puffed and golden and the edges pull away from the sides of the pan, about 20 to 25 minutes.
Meanwhile, place the three milks and the rum in a large bowl and whisk until combined; set aside.
Remove the cake from the oven and place on a wire cooling rack.
Using a toothpick or wooden skewer, poke holes all over the cake and allow to cool for 15 minutes.
Pour the milk mixture evenly over the cake and continue cooling, about 45 minutes more.
Tightly cover with plastic wrap and refrigerate at least 4 hours or overnight.
When the cake is ready to serve, spread the coconut in an even layer in a large frying pan.
Toast over medium heat, stirring often, until lightly browned and fragrant, about 5 minutes. (If the coconut begins to burn, reduce the heat.)
Immediately remove from the pan to a small bowl.
Place the heavy cream and powdered sugar in large bowl and whisk until medium peaks form. (If you like, flavor it with a teaspoon of dark rum.)
Slice the cake and serve topped with a mound of whipped cream and a sprinkle of toasted coconut.