Combine 1 cup butter, sugar and egg in bowl. Beat at medium speed, until creamy. Add milk, 1 1/2 teaspoons vanilla and peppermint extract. Continue beating, scraping bowl often, until well mixed. Add flour, baking powder and peppermint candies; beat at low speed, scraping bowl often, until well mixed.
Divide dough into thirds. Shape each third into a ball; flatten to 1/2 inch. Wrap each in plastic food wrap. Refrigerate 2-3 hours or until firm.
Heat oven to 400°F.
Roll out dough on lightly floured surface, one-third at a time (keeping remaining dough refrigerated), to 1/8 to 1/4-inch thickness. Cut with cookie cutters. Place 1 inch apart on ungreased cookie sheets. Bake 6-10 minutes or until edges are lightly browned. Cool completely.
Combine powdered sugar, water, 1 tablespoon butter, corn syrup and 1/2 teaspoon vanilla in bowl; mix until powdered sugar is moistened. Beat at medium speed until smooth, adding additional water if necessary, to reach desired glazing consistency. Tint with food color, if desired.
Glaze cooled cookies. Add decorator candies, if desired. Let stand 6 hours or overnight until hardened.
*Substitute peppermint extract.