In a large mixing bowl, beat the 2/3 cup butter and shortening with an electric mixer on medium to high speed for 30 seconds. Add the 1-1/2 cups sugar, baking powder, and salt. Beat until combined, scraping sides of bowl occasionally. Add the eggs and peppermint extract; beat until well combined. Beat or stir in the 4 cups flour. Using a wooden spoon, stir in the peppermint candies.
Divide dough in half. Cover and chill about 3 hours or until dough is easy to handle.
Meanwhile, in a medium mixing bowl beat 1/2 cup butter with an electric mixer on medium speed for 30 seconds. Add 1/2 cup sugar and beat until combined. With mixer on low speed, beat in half-and-half or light cream. Beat in 2 cups all-purpose flour until dough is smooth. Divide dough into portions. Tint dough with desired paste food coloring. Set aside.
On a lightly floured surface, roll half of the chilled dough at a time to 1/4-inch thickness. Using a 4-inch round scalloped cookie cutter, cut out dough. Place cookies about 1 inch apart on an ungreased cookie sheet. Tuck a lollipop stick under the center of each cookie. Press dough down slightly so that the cookie bakes around the stick.
Using a decorative bag fitted with a medium writing or star tip, pipe colored dough onto cutouts, making borders, designs, or pictures and/or writing messages.
Bake in a 375 degree F oven for 8 to 10 minutes or until edges are firm and bottoms are lightly browned. Cool on cookie sheet for 1 minute. Carefully transfer cookies to a wire rack; cool. Makes about 24 (4-inch) cookies.