Add the cake mix, vegetable oil, and eggs to a large bowl or stand mixer fitted with a paddle attachment and beat until combined.
Divide the dough evenly into the number of different colors you want, putting each portion into its own bowl. (In this recipe, we did 5.)
Mix each portion of dough with the desired color using a hand mixer for even coloring and the easiest mixing method. (Be sure to wash the beaters before moving on to the next bowl.)
Chill the dough for 45 minutes. (see note)
Preheat your oven to 350 degrees F and line a baking sheet with parchment paper.
Use two teaspoons to portion out the dough into dollops, using the spoons to work the dough back and forth between them until the dough drops onto the parchment. (The dough will be very sticky.) Drop one dollop of each color of dough per cookie.
Use a medium cookie scoop to scoop the colored dollops together. Gently roll them into balls, you want to avoid getting food coloring all over your hands. Then roll the dough balls in the sugar to coat. Place the cookie dough balls on the prepared baking sheet at least 2 inches apart.
Bake for 8 to 9 minutes. Do not overbake these cookies, they will look puffy and slightly underdone, but they will finish cooking after coming out of the oven.
Let the cookies cool on the baking sheet for about 5 minutes before transferring to a wire rack to cool completely.
Notes
It is extremely important that you chill these cookies for at least 45 minutes and not longer than 1 hour. The dough will be too soft to shape and roll without chilling. Prior to the 1-hour mark, these cookies really puff up and have more of a soft and chewy texture. After 1 hour they can become hard and crunchy.
I used Wilton Color Right food coloring, but you could also use regular food coloring or gel coloring. These will have varying strengths and certain colors may transfer off the other dough more than others. You can do anywhere from 2 to 6 colors with this method and the colors can be customized to match your celebration.
If you don't want to use a cookie scoop, or don't have one, you can pop on some food-safe gloves to help protect your hands from the food coloring.
Store in an airtight container for up to 1 week or freeze for up to 3 months.