In a mixing bowl, beat butter with an electric mixer on low speed or by hand until creamy, then blend in egg. Slowly add dry cookie mix and blend until a dough is formed. Divide dough into three equal portions.
Return one third of dough to the mixing bowl. Add red natural food dye and mix until dough is evenly colored. Repeat with another portion of uncolored dough and the green food dye.
Place remaining uncolored dye between two large sheets of parchment paper. Roll into a rectangle shape until dough is between 1/8” - 1/4” thick. Place dough into refrigerator to chill. Repeat with red and green dyed dough. Chill dough 10 minutes, or until firm.
Once chilled, place the red colored dough down first and remove the top layer of parchment paper. Peel away a layer of parchment paper from the uncolored dough, then place on top of the red dough and remove remaining parchment paper. Finally, unwrap the green dough and place on top of the uncolored dough.
Trim dough into an even rectangle. Starting with one of the short ends, tightly roll the dough into a log, using the bottom layer of parchment to help roll the dough. Wrap the dough in plastic wrap and chill at least 30 minutes or until completely firm. Place dough inside of two large drinking glasses to help keep its rounded shape.
Use a sharp knife to cut off the ends of the dough. Combine sprinkles and spread onto a shallow baking dish or tray. Unwrap dough and roll entirely in the sprinkles, pressing sprinkles into the sides until the dough is completely covered.
Preheat oven to 350 degrees and line a baking sheet with parchment paper. Slice 1/2” thick cookies and place at least 2” apart on prepared pan. Bake 12-15 minutes, until the edges begin to turn golden and the middle is set. Allow cookies to sit on the pan 5 minutes before transferring to a wire rack to cool completely. Enjoy!