Preheat the oven to 350°F (175°C). Lightly grease a 9×13-inch baking pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, beat the softened unsalted butter and granulated sugar together on medium speed until light and fluffy, about 3 minutes.
Add the egg and vanilla extract to the butter mixture and beat until combined.
Gradually add the dry ingredients to the wet ingredients in three additions, alternating with the whole milk, beginning and ending with the dry ingredients. Mix until just combined after each addition.
Spread the cookie dough evenly into the prepared baking pan, smoothing the top with a spatula.
Bake in the preheated oven for 23 to 25 minutes, or until the edges are lightly golden and a toothpick inserted into the center comes out clean.
Remove from the oven and allow the bars to cool completely in the pan on a wire rack.
Once cooled, cut into 16 squares and serve.