Cookies: Preheat oven to 350 degrees. Butter a sheet pan. In a mixer fitted with a paddle attachment (or using a hand mixer), cream the sugar, butter and shortening until fluffy. Add the egg, vanilla and baking powder and mix. Add the flour and mix. Shape the dough into a large flat disk, kneading briefly if necessary to bring the dough together. Wrap in plastic wrap and chill 1 to 2 hours.
On a lightly floured surface, roll out the dough 1/4-inch thick. Using cookie cutters or empty tuna cans, cut out large round cookies about 3 inches in diameter. Transfer to the prepared pan. Cut a heart or circle out of the center of each cookie that will be filled later with lollipop syrup. Carefully slip the sucker sticks under the cookies and gently press dough down and around it to secure the stick. Bake until very light golden, 20 to 25 minutes. Let cool completely on the sheet pan.
Lollipop Centers: Combine sugar, glucose and water in a clean, dry, small saucepan fitted with a candy thermometer and cook on high heat. While cooking the candy syrup, occasionally wash down the pan sides to prevent crystallization using a clean brush dipped in water. Without stirring, cook until mixture reaches 305 degrees F or hard crack on the candy thermometer. Remove the pot from the heat and dip it into an ice bath for 10 seconds to stop the cooking. Remove the pot from the ice bath and add food coloring, stirring well a wooden spoon so the color is evenly distributed. To avoid air bubbles, stir in both directions and be careful not to over mix. Pour the syrup into hole of the cookies. Cool at least 20 minutes to harden. Store in an airtight container.