In a large bowl, cream butter and sugar until light and fluffy. Beat in egg and vanilla. Combine the flour, nutmeg and salt; gradually add to the creamed mixture. Cover and refrigerate for 1 hour or until firm.
On a sugared surface, shape 1/2 cupfuls of dough into 1/2-in.-thick logs. Cut logs into 2- to 2-1/2-in. pieces.
Place 2 in. apart on ungreased baking sheets. Bake at 350° for 12-14 minutes or until lightly browned. Remove to wire racks to cool.