Beat butter at medium speed with an electric mixer until creamy; gradually add sugar, beating well. Add eggs, beating until blended.
Combine flour, baking powder, and nutmeg; add to butter mixture alternately with milk, beginning and ending with flour mixture. Beat at low speed just until blended after each addition.
Divide dough in half; cover with plastic wrap, and chill 1 hour.
Roll half of dough to 1/4-inch thickness on a lightly floured surface. Cut with a 2 1/4-inch round cutter, and place on greased baking sheets. Repeat procedure with remaining half of dough.
Bake at 350° for 8 minutes. (Cookies will be pale.) Cool on wire racks.