In a mixing bowl, cream together the butter and powdered sugar.
Mix in the egg and vanilla. Thoroughly combine.
In another bowl, sift together the flour, cocoa powder, baking soda, salt, and cream of tartar.
Add the dry ingredients in batches to the creamed mixture until you have combined everything well.
Cover and refrigerate the dough for at least 30 minutes. This is to harden up the fat so the cookies don't spread too much.
Preheat the oven to 350 degrees
Cut the dough in half.
Roll one half out onto a floured surface to around a quarter-inch thickness. Cut the cookies out with a Shamrock cookie cutter
Transfer cookies to a parchment or silicone lined half sheet pan. Do no more than 6 cookies on a pan.
Bake for 10-14 minutes or until slightly browned around the edges
Allow to cool on a wire rack before frosting.
To make the buttercream
Whip the butter until it's fluffy.
Incorporate the sugar 1 cup at a time.
Add the oil and whip frosting until light and fluffy.
Add milk as needed to get desired consistency
Notes
I prefer using paste food coloring. It has more vivid colors and doesn't add moisture to the dough. You can find it at a good baking store or online. Use regular food color if you like.