Preheat the oven to 325 degrees F and line a baking sheet with parchment paper. Heat the milk, 1/4 cup sugar, the nutmeg and a pinch of salt in a small saucepan over medium-low heat, stirring to dissolve the sugar.
Whisk the egg yolks and cornstarch in a medium bowl. When the milk mixture starts to steam, slowly pour half the milk into the egg yolks, whisking constantly. Pour the egg mixture into the saucepan with the remaining milk and cook, whisking occasionally, until thick and spoonable, about 3 minutes. Remove from the heat and whisk in 1/4 teaspoon vanilla. Transfer the custard to a bowl and let cool to room temperature.
Meanwhile, whisk the egg whites, rum, remaining 1/2 cup sugar, 1 teaspoon vanilla and 1/4 teaspoon salt in a large bowl. Fold in the coconut. Cover and refrigerate about 30 minutes to firm up.
Scoop level tablespoonfuls of the coconut mixture about 1 inch apart on the prepared pan. Make an indentation in the middle of each cookie with the back of a 1/4-teaspoon measuring spoon. Fill each indentation with the cooled custard.
Bake until the cookies are golden around the edges, 20 to 25 minutes. Grate some more nutmeg on top of the cookies. Let cool 10 minutes on the pan, then transfer to a rack to cool completely.