Preheat the oven to 350°. Sift together the flour, baking powder, salt, cinnamon and nutmeg; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and sugar together until light and fluffy, about 2 minutes. Add egg yolks, one at a time, beating after each addition. Add the eggnog, vanilla and rum; beat until combined.
Slowly add the flour mixture, beating on low until just combined.
Drop by rounded tablespoons onto silpat- or parchment-lined baking sheets, about 2 inches apart. Bake until light golden on the bottoms, about 10-12 minutes. Transfer to a cooling rack until ready to frost.
Meanwhile, make the icing. In the bowl of an electric mixer fitted with a paddle attachment, beat the butter and cream cheese together until well combined. Add the powdered sugar, and beat slowly until incorporated. Add rum, lemon juice and a dash of salt; beat on medium-high until the icing looks fluffy and spreadable.
Top each cookie with about 1 1/2 teaspoons of the icing. If desired, grate a little bit of fresh nutmeg over the top.