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Beat 2/3 cup butter in a mixing bowl with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar and 1/8 teaspoon nutmeg; beat until combined. Beat in egg yolks and vanilla. Beat in as much flour as you can. Stir in any remaining flour with a wooden spoon. If necessary, cover and chill dough about 1 hour or until easy to handle.
Preheat oven to 375°F. Shape dough into 1-inch balls. Roll balls in egg whites, then in chopped walnuts. Place 1 inch apart on lightly greased baking sheets. Press centers with your thumb. Bake for 10 to 12 minutes or until edges are lightly browned. Cool on wire racks.
For filling, beat 1/4 cup butter until softened. Add confectioners' sugar and beat until fluffy. Beat in rum (or rum extract) and enough milk to reach spreading consistency. Spoon or pipe about 1/2 teaspoon filling into the center of each cookie. Sprinkle with nutmeg.