Beat the butter and sugar in a large bowl with a mixer on medium-high speed until light and fluffy, 3 to 5 minutes. Beat in the egg and vanilla. Reduce the mixer speed to low; add the flour mixture in 2 batches and beat until just incorporated. Divide the dough in half; wrap in plastic wrap and refrigerate until firm, at least 1 hour and up to 1 day.
Meanwhile, separate the candy by color and place each color in a separate resealable plastic bag. Crush into small pieces using a rolling pin; do not pulverize.
Line 2 baking sheets with parchment paper. Working with 1 piece of dough at a time, lightly dust the dough with flour and roll out between 2 pieces of parchment paper until about 1/8 inch thick. Refrigerate until slightly firm, about 15 minutes. Cut into shapes using 2-to-4-inch cookie cutters and transfer to the prepared baking sheets. Gather the scraps and refrigerate until firm; reroll once to cut out more cookies. Cut out shapes from the centers of the cookies using smaller cutters; fill the cut-out areas about two-thirds full with the crushed candy. Refrigerate until firm, about 30 minutes.
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Bake the cookies, switching the position of the pans halfway through, until the candy melts and the cookies are just golden, 12 to 15 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.