Preheat the oven to 350ºF. Line a baking sheet with parchment paper or a Silpat. Set aside.
In a medium bowl combine flour and baking powder. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and sugar on low speed. Add the egg and vanilla and mix. Gradually add the flour mixture and beat just until combined, scraping down the bowl as needed.
The dough should look crumbly. Turn it out onto a clean work surface and knead until soft and smooth.
Roll the dough on a floured work surface to about ¼" thick and cut into shapes. You can take the scraps and gently form them into another ball to roll and cut again, but be careful not to add too much flour or work the dough a lot or your cookies will become tough and dry.
Place the cut out cookies on the prepared baking sheet and bake 10-12 minutes. Let cool a few minutes on the sheet, then transfer to a cooling rack to cool completely before frosting.
Vanilla Buttercream Frosting
In a large bowl combine butter and powdered sugar using a hand mixer on low speed.
Add vanilla and mix again on low.
Scrape down the bowl and add the milk a few drops at a time until desired consistency is reached, mixing on medium low and scraping the bowl occasionally.
Use immediately or divide and tint, then use to frost cooled cookies.
Video
Notes
Tips for customizing Christmas cut out cookies
I've made them dairy free with great success by using a vegan buttery spread and almond milk
Make them gluten-free by using a gluten-free flour blend
Add extra flavor to the frosting by swapping the vanilla extract for peppermint extract, almond extract, or cinnamon (you may need to adjust quantities - add a little to start with and add more as necessary)
Add a touch of nutmeg to the cookie dough for a lightly spiced cookie