Make the Cookie Bars: Preheat oven to 350 degrees F. Grease a 9x13-inch baking pan or line with parchment paper; set aside.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt; set aside.
Using an electric mixer, cream together the butter and cream cheese on medium speed until smooth, about 1 minute. Add the sugar and continue to beat until light and fluffy, about 3 minutes. Add the egg and vanilla extract and beat to combine, about 1 minute. Reduce the mixer speed to low and gradually add the flour mixture, mixing only until barely combined. Use a rubber spatula to finish mixing, ensuring there are no pockets of flour remaining.
Using a spatula, spread the dough into an even layer in the prepared pan (the dough will be thick!). Bake until light golden brown and a toothpick inserted in the center comes out clean, 20 to 25 minutes. Place the pan on a wire rack to cool completely before frosting.
Make the Vanilla Frosting: Using the wire whisk attachment of your stand mixer, whip the butter on medium-high speed for 5 minutes, stopping to scrape the bowl once or twice. Reduce the speed to low and gradually add the powdered sugar. Once all of the powdered sugar is incorporated, increase the speed to medium-high and add the vanilla, mixing until incorporated. Whip at medium-high speed until light and fluffy, about 2 minutes, scraping the bowl as needed.
Spread the frosting over the cookie bars and decorate with sprinkles, if desired. Cut the bars into squares and serve. The cookie bars should be kept in an airtight container at room temperature for up to 3 days (you can store them in the refrigerator for up to 1 week).