Open the three rolls of sugar cookie dough and divide the dough into four equal portions. Tint one portion red, one yellow, one orange (by combining red and yellow), and one turquoise (by combining blue and green).

Divide each colored dough portion into three smaller pieces. Measure the height of your burro-shaped cookie cutter (or another desired shape), then find a container that matches the cutter's height to use as a mold. Layer the dough colors alternately in the container in your preferred order. Cover and refrigerate until the dough is firm.

Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper. Remove the layered dough from the container and slice into ¼-inch thick pieces. Place slices on the prepared baking sheets, spacing them 2 inches apart.

Bake for 6 to 10 minutes, or until the cookies are set. While still warm, use your burro-shaped cookie cutter to cut out shapes from the baked slices. For 16 of the cookies, use a small round cutter to cut out the centers, creating windows for the candy filling. Carefully remove the center cutouts. Allow cookies to cool for 5 minutes on the baking sheet, then transfer to a wire rack to cool completely.

In a small bowl, whisk together the powdered sugar, milk, corn syrup, and vanilla until smooth. Add food coloring of your choice to tint the icing. Transfer the icing to a piping bag fitted with a small round tip or a resealable plastic bag with a small corner snipped off.

To assemble each piñata sandwich cookie, pipe a ring of icing around the edge of one whole burro-shaped cookie. Place a cutout cookie on top and pipe another ring of icing around the edge. Fill the hollow center with candy-coated chocolate candies. Finally, top with another whole burro cookie, pressing gently to seal.