Preheat your oven to 375 degrees Fahrenheit and grease a 9-inch by 13-inch rectangular baking pan with butter or baking spray. You can also line the baking pan with parchment paper for easy removal.
In a large bowl using a hand mixer, or in the bowl of your stand mixer fitted with the paddle attachment, beat the butter and sugar on high speed for several minutes until they're creamed together (fluffy and smooth and pale in colour).
Beat in the sour cream, egg, and vanilla extract until well incorporated. Scrape down the sides of the bowl with your spatula to make sure everything is incorporated evenly.
Add the flour, baking powder, and salt and mix on low speed just until combined. The mixture may appear slightly crumbly but this is normal.
Press the mixture into the bottom of the prepared baking pan with your hands or a flat kitchen object like the bottom of a measuring cup. Even out the top as much as possible.
Bake the sugar cookie base at 375 degrees Fahrenheit for about 15-17 minutes, making sure that no browning occurs on the edges. If the edges start to brown, remove the base from the oven immediately.
Let the sugar cookie base cool completely to room temperature while you make the frosting.