Preheat the oven to 350°F (175°C). 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, cream the softened unsalted butter and granulated sugar together until light and fluffy, about 3 to 4 minutes.
Add the egg and vanilla extract to the butter mixture and beat until fully combined.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined.
Pour in the whole milk and mix until the dough is smooth and well combined.
Spread the dough evenly into the prepared baking pan, smoothing the top with a spatula.
Bake in the preheated oven for 22 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 before cutting into 16 squares.