Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Spread the walnuts on another baking sheet and bake until toasted, about 10 minutes. Let cool, then pulse the nuts in a food processor until finely ground. Transfer to a small bowl and add the rum and vanilla; let soak at least 30 minutes.
Whisk the flour, 1 teaspoon nutmeg and the salt in a medium bowl. Beat the butter, egg yolk and 1/2 cup confectioners' sugar in a large bowl with a mixer on medium-high speed until creamy, about 3 minutes. Beat in the walnut-rum mixture until fluffy, about 1 minute. Reduce the mixer speed to low and beat in the flour mixture, then increase the speed to medium high and beat until combined. Turn out the dough onto a large sheet of plastic wrap and pat into a disk; wrap and refrigerate until firm, about 1 hour.
Roll tablespoonfuls of dough into 1-inch balls and arrange 1 inch apart on the prepared pans. Bake until light golden, 20 to 25 minutes. Let cool 5 minutes on the pans.
Put the remaining 1 cup confectioners' sugar and 3/4 teaspoon nutmeg in a medium bowl. Gently roll the warm cookies in the confectioners' sugar mixture. Transfer to a rack to cool completely; the cookies will firm up as they cool. Once cool, reroll in the confectioners' sugar mixture.