Beat butter in large bowl with electric mixer on medium speed 30 seconds or until smooth. Add sugar, eggs and vanilla extract; beat until well blended.
Gradually beat in flour and nutmeg, scraping side of bowl occasionally. Stir in nuts. If needed, refrigerate 30 minutes or until dough is easy to handle.
Shape dough into 1-inch balls. Place about 1 inch apart on baking sheets. (Refrigerate remainder of dough while baking each batch of cookies.)
Bake in preheated 375°F oven 10 minutes or until edges are lightly browned. Remove from baking sheets. Cool completely on wire racks.
Prepare frosting. Beat butter in small bowl until softened. Gradually add sifted confectioners' sugar; beat until fluffy. Beat in milk and imitation rum extract until well blended. Frost center of each cookie with about 1 teaspoon frosting. Sprinkle with additional nutmeg, if desired.