Whisk together the flour, baking soda, baking powder and salt until well combined. Set aside.
In a separate bowl, beat together the butter, cream cheese, and sugar until light and fluffy.
Add the egg and vanilla and beat to combine.
Mix in the flour mixture until a thick dough forms.
Work the green food coloring into the dough using your hands. Wear plastic gloves to keep the coloring from staining your hands.
Separate the dough into two balls, then flatten the balls into disks. Wrap each disk in plastic wrap and place in the refrigerator to chill for at least one hour.
When ready to bake, preheat the oven to 375 degrees and line a baking sheet with parchment paper.
Roll out the dough on a lightly floured surface until it’s around 1/8th inch thick, then use shamrock cookie cutters to cut out the cookies. After cutting out a set, combine all the scraps and re-roll the dough until all the cookies have been cut.
Bake the cookies for 8 to 10 minutes, until the edges are golden brown. Allow the cookies to cool on the baking sheet for five minutes, then transfer to a wire cooling rack to cool completely.