Beat butter and sugar in large bowl until smooth and creamy. Beat in egg. Beat in peppermint and vanilla extracts. On low speed, beat in flour mixture. Shape dough into a ball; wrap in plastic wrap. Refrigerate 1 hour.
Heta oven to 375. Coat baking sheet with non-stick cooking spray.
Shape dough into 1 inch balls for a total of 36 cookies; place on prepared baking sheets.
Bake at 375 for 10 to 12 minutes or until bottoms are lightly browned. Transfer cookies to wire racks to cool completely. When cool, brush with melted white chocolate; dip in chopped candy canes. Let harden on wax paper.