Add the room temperature butter to a bowl or bowl of a stand mixer. Add in the sugar and cream together for about 30 seconds, or until well combined.
Beat in the egg and vanilla.
Add in all remaining ingredients and using a wooden spoon or rubber spatula to combine.
Once a dough has formed, add it right into the fridge for at least 1 hour, ideally 2, but up to 24 hours. I know chilling dough is annoying, but please don't skip this step :)
Once the dough is ready, preheat the oven to 350 degrees Fahrenheit.
Add the dough to a piece of parchment paper and roll with a rolling pin until it is about ¼ inch thick. If you prefer thicker cookies, then feel free to keep it thicker!
Use a cookie cutter (I used these round ones) and cut the dough. I find it easiest to slide a lightly greased spatula under the cookie with the cookie cutter still on, then add it to a baking sheet lined with parchment paper, slide away the spatula, then lift off the cookie cutter so it keeps its shape.
Repeat until all dough has been used, rolling it out again as needed. The cookies will spread very slightly while baking, so keep a bit of room between each cookie. Only bake one cookie sheet at a time, so do two batches if needed.
Bake for 9-12 minutes. I usually do 11 in my oven, but all ovens are different. You'll know they're done when the edges are just slightly golden brown. They go from golden to burnt pretty quickly, so keep an eye on them! They'll be soft when they come out of the oven, but will continue to firm up as they cool.
To make the icing, add the room temperature butter to a bowl and beat with an electric mixer until creamy. Add in the powdered sugar and beat until well combined. Finally, mix in the vanilla.
Wait until the cookies are completely cool, then frost and enjoy!