In a mixing bowl, beat butter with electric mixer on low speed until creamy, then blend in egg. Slowly add in sugar cookie mix and blend until incorporated. Gather dough in a ball and flatten into a disc. Wrap dough in plastic and freeze for 20 minutes, until dough is firm.
Preheat oven to 350°F. Remove dough from freezer. Place dough between two sheets of parchment paper and roll to about 1/4" thick. Cut with desired holiday cookie cutters. Place cookies 1" apart on parchment-lined baking sheet. Bake for 12-14 minutes until edges lightly browned. Remove and let cool for 5 minutes.
While cookies cool, prepare the royal icing. In a mixing bowl, whisk egg whites and vanilla extract with an electric mixer on low speed, until foamy. Slowly add in powdered sugar until the shiny and stiff peaks form.
Divide icing into separate bowls for desired icing colors. Mix in a pea-sized amount of gel food dye until blended. Pour each color into separate piping bags, fitted with a #2 icing tip. Create an icing outline on the sugar cookies first. Let set for about 5 minutes, then flood the middle with icing until completely filled and smooth. Add any desired sprinkles or accessories while icing is wet. Let icing set, then add additional frosting decorations and details. Serve and enjoy!