In a mixing bowl. beat butter with electric mixer on low speed until creamy, then blend in egg. Slowly add in dry cookie mix and blend until a dough forms.
Divide the dough into six equal portions. Mix 1/2 teaspoon of each color dye into the dough, blending thoroughly.
Roll each portion of dough in between two sheets of parchment paper to form a rectangle, about 1/8" thick. Freeze the dough sheets for 10 minutes.
Remove dough from freezer and remove the top sheet of parchment from the red dough. Remove parchment from orange dough sheet and stack on top of the red dough. Remove parchment from remaining dough sheets and stack in the order of the colors of the rainbow, ending with the violet dough.
Roll the dough along the longer side like a jellyroll, with the red colored dough on the top. Cover with plastic and freeze 15 minutes.
Preheat oven to 350 degrees and line a baking sheet with parchment paper. Remove dough from freezer and unwrap from plastic. Slice the dough into 1/4" thick round cookies. Place on parchment, at least 1 inch apart. Bake 12-15 minutes until slightly puffy and golden.