In a large bowl, with an electric mixer on medium speed, beat butter and sugar until blended, then beat on high speed until creamy. Add egg and vanilla and beat until well blended.
Add flour and salt; beat on low speed until combined, then on medium speed until well blended. Divide dough into three equal portions.
Place each portion between sheets of floured plastic wrap. Press evenly into a disk about 1/4 inch thick. Stack disks on a baking sheet and chill until dough is firm, at least 15 minutes, or up to 2 days.
Working with one portion of dough at a time (keep remaining chilled), peel off plastic wrap and set disk on a floured board. With a floured rolling pin, roll dough about 1/8 inch thick. With floured cookie cutters (2 to 4 in. wide), cut out cookies, placing cutters as close together as possible. Transfer to baking parchment-lined or bare baking sheets (don't butter sheets), placing cookies about 1 inch apart. Gather up dough scraps and press together into a ball. If dough is still cold, roll and cut out remaining cookies. If dough is soft and sticky, press scraps into a ball and repeat step
Bake cookies in a 350° oven (325° for convection) until golden around edges, 7 to 9 minutes. If baking more than one sheet at a time in one oven, switch positions halfway through baking. Slide parchment with cookies off sheets and onto racks or a counter to cool, then remove from parchment. Or cool on sheets about 1 minute, then, with a wide spatula, transfer cookies to racks to cool completely. If not using baking parchment, wipe off baking sheets after each batch. Cool sheets before filling with more cookies.
To decorate: Gather royal icing, cake-decorating sugar, jimmies, colored sprinkles, miniature chocolate chips, and/or red-hot candies.
Fold down the top edges of a 1-quart heavy zip-lock plastic bag to form a cuff about 3 inches wide.
Hold the bag under cuff and scoop in 1 to 2 cups icing. Squeeze the icing down into one corner of the bag. Squeeze all the air out of bag, and seal.
When ready to use, cut about 1/16 inch off the filled corner of the plastic bag. Twist bag just above icing. Holding twisted end with one hand, gently press bag from the top to pipe icing onto cookie. Twist bag tighter as it empties. Sprinkle or gently press decorative sugar, candies, or chocolate chips into soft icing.
When you're not using the icing bag, fold the opening closed and seal with a paper clip.
Let cookies stand until icing is firm, 30 to 45 minutes. Store iced cookies airtight up to 3 days.