Preheat oven to 350°F. Line an 8x8-inch baking pan with parchment paper.
In a large mixing bowl with an electric mixer, combine cookie mix, butter, and egg until a soft dough is formed.
If using, fold in sprinkles and mix until they are evenly incorporated.
Spread cookie dough into prepared pan.
Bake in preheated oven for 23-25 minutes.
Remove pan to wire rack and cool completely.
Once the cookies are cool, prepare the buttercream.
For the Buttercream
In the bowl of a stand mixer fitted with the paddle attachment, or in a large mixing bowl with an electric mixer, cream butter on medium-high speed for about 6 minutes. With the mixer on low speed, slowly add in powdered sugar, and continue beating until well blended.
Add in vanilla, and 1 tablespoon of milk. Blend on low speed until moistened.
If needed, add additional milk 1 teaspoon at a time, until you reach the desired consistency. Beat at high speed until frosting is smooth and fluffy; about 5 more minutes.
If desired, use gel food dye to color the frosting.
Spread frosting onto cookie bars, or use a small biscuit cutter to cut the cookie bars into small circles. Pipe frosting onto cookies and top with festive sprinkles.
Notes
This buttercream recipe makes a generous amount of frosting. If you do not desire to pipe the frosting onto the cookies, I would suggest cutting the buttercream recipe in half.