Preheat the oven to 350°F (175°C). Line a 9×13-inch baking pan with parchment paper, leaving an overhang on the sides for easy removal.
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 1 1/4 cups 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 fully combined.
Gradually add the dry ingredients to the wet ingredients, alternating with the whole milk, beginning and ending with the dry ingredients. Mix until just combined, being careful not to overmix.
Spread the dough evenly into the prepared baking pan and smooth the top with a spatula.
Sprinkle the 1/4 cup granulated sugar evenly over the top of the dough.
Bake in the preheated oven for 22 to 25 minutes, or until the edges are lightly golden and a toothpick inserted in 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, use the parchment paper overhang to lift the bars out of the pan and cut into 16 squares.