Combine 1 cup butter, 1 cup sugar, 1 egg, 1/2 tsp vanilla, and 1/2 tsp peppermint extract in mixing bowl. Beat at medium-high speed until creamy, scraping bowl often.
Reduce speed to low. Add 2 cups flour, 1/2 cup unsweetened cocoa, 1/2 tsp baking soda, and 1/4 tsp salt. Mix well.
Wrap dough in wax paper or plastic food wrap. Refrigerate until firm (1 hour or overnight).
Heat oven to 400 degrees. Roll out dough on lightly floured surface to 1/8 inch thickness. Cut with cookie cutters. Place 1 inch apart on ungreased cookie sheets. Bake 5-6 minutes, or until set. Let stand 1 minute, then remove to wire racks to cool completely.
Frost and decorate cookies with a Peppermint Royal or Buttercream Frosting.