Using electric mixer, beat butter and sugar in large bowl until fluffy. Beat in egg and vanilla. Sift flour, baking powder, salt and nutmeg over; stir to blend well. Turn dough out onto lightly floured surface and knead gently 1 minute. Shape dough into 1/2-inch-thick rectangle. Cut into 4 equal pieces; wrap in plastic and refrigerate at least 3 hours and up to 1 day. Let dough soften slightly at room temperature before rolling out.
Position rack in center of oven and preheat to 350°F. Butter large baking sheet. Working with 1 dough piece at a time, roll out dough on lightly floured surface to 1/8-inch thickness, lifting and turning dough often and dusting surface very lightly with flour to prevent sticking. Using floured 3- to 4-inch cutters, cut out cookies. Pull away excess dough from around cookies. Transfer cookies to prepared baking sheet, spacing 1 inch apart (cookies will not spread). If using cookies as hanging ornaments, push 1 end of drinking straw through dough near top of each cookie; lift straw, then remove small dough round from straw. Gently reroll dough scraps; cut out more cookies. Transfer to same sheet.
Bake cookies until light brown, about 11 minutes. Let cool 5 minutes on sheet. Transfer cookies to rack; cool.
Repeat with remaining dough pieces, baking 1 sheet of cookies at a time. Cool baking sheet completely and butter sheet lightly between batches
