In the bowl of a stand mixer fitted with the paddle attachment, or using a hand mixer, cream the butter and sugar for 2 minutes on medium speed. Add eggs and vanilla. Mix until combined. Add the flour and baking powder. Mix until combined.
Preheat oven to 350°F.
Roll out dough on a floured surface (keep dough about 1/4" thick, or thicker if that's what you prefer). Use cookie cutters to cut into your desired shapes. Place on a parchment lined baking sheet.
Place the baking sheet in the freeze for 10 minutes.
Bake at 350°F for 10 minutes. Remove from oven and let cool on pan for 5 minutes. Transfer to a wire rack to cool completely before frosting. Can frost with royal icing, store-bought icing or homemade frosting.
Frosting
Add all the ingredients to a medium sized bowl and beat together until combined. Add additional milk or powdered sugar to reach your desired frosting consistency.
Video
Notes
If you want to use unsalted butter that's fine, just add 1/4 tsp salt when you add the flour and baking powder to the dough. Storing and Freezing: Store the cookies frosted or unfrosted in an airtight container or Ziplock bag at room temperature for 4-5 days or in the fridge for up to 10 days.Freeze plain or iced cookies in the freezer for up to 3 months. If iced, wait until the icing has set before layering with parchment paper and storing in the freezer in an airtight container or Ziplock bag. If frosted, flash freeze the cookies on a baking sheet until the frosting is set/hard. Then store layered with parchment paper in an airtight container or Ziplock bag in the freezer.Remove from the freezer and let thaw at room temperature or in the fridge.Because these cookies will stay really well in the fridge or freezer, they are great make ahead sugar cookies! You can also store the sugar cookie dough in the fridge for 1-2 days prior to rolling, cutting and baking or you can freeze the cookie dough in the freezer in an airtight container or Ziplock bag for 1-2 months prior to rolling out and baking.