In a medium saucepan, melt butter. Stir in sugar. Remove from heat. Cool for 15 minutes.
With a spoon, beat in eggs and vanilla. Place in a large bowl. Stir in flour, baking powder, salt, and nutmeg. Cover surface with clear plastic wrap. Chill in the refrigerator about 4 hours or until easy to handle.
On floured surface, roll half of dough to 1/8- to 1/4-inch thickness. Cut into shapes with cookie cutters. Place on an ungreased cookie sheet. Bake in a 375 degree F oven for 6 to 8 minutes or until edges are very lightly browned. Cool cookies on wire rack.
Frost with Paintbrush Icing. Makes about 60 (3-inch) cookies.
In a medium bowl, combine powdered sugar and meringue powder.* Beating with an electric mixer on low speed, gradually add water. Beat on high speed for 2 to 3 minutes or until icing becomes thick. Beat in lemon extract. Divide icing into smaller portions for different colors. Tint icing with paste or liquid food coloring. (Keep icings covered with plastic wrap to prevent drying out.) Thin colored icings with additional water until the consistency of thick syrup. Spoon a small amount of icing onto a cookie to be decorated. Spread icing with a metal spatula or use clean artists paintbrushes to paint it onto the cookies in desired design.
Look for meringue powder for the frosting at stores selling cake-decorating and crafts supplies.