Place same colors of candies in separate zip-top bags. Use a rolling pin or small hammer to crush the candies. Pour each color of crushed candy into separate bowls.
Beat butter and sugar together in the bowl of a stand mixer just until incorporated. Add egg and almond extract. Mix again on low speed. Scrape down the bowl as needed.
Add the flour and salt. Mix on low speed until dough is formed and there are no longer any streaks of butter in the mixing bowl.
Line two baking sheets with parchment paper. Turn the dough out onto a sheet of plastic wrap and form into a ball. Wrap the dough tightly and refrigerate 1 hour.
Preheat oven to 350°F.
Working on a well-floured surface, knead dough slightly, squeezing it with your hands to flatten ball into a disc. Roll dough to 1/4-inch thickness. Cut shapes with larger (4-inch) cookie cutters, and then stamp a smaller shape from the center of the larger cookie. Lay shapes with cut-outs in the centers on a cookie sheet. Refrigerate 30 minutes. This will help the cookies maintain a crisp shape during baking.
Use a spoon to fill the centers of cookies with crushed candies. Bake 5-7 minutes, or until crushed candy is melted. Remove cookies from oven and carefully fill each with more crushed candy. Bake an additional 5 minutes, or until the newly added crushed candy is melted and smooth. Remove pan from oven and let cookies cool completely on pan. When the cookies are cool, gently remove them to a platter and serve.