Preheat oven to 350°F. Line a lightly greased (with cooking spray) 9-inch square pan with parchment paper.
Cream the butter and 1 1/4 cups sugar together in a large bowl (either of a stand mixer or just a bowl with a hand mixer) until light and fluffy. I like to do this for about 2 to 3 minutes.
Beat in the eggs and egg yolks, one at a time, until completely incorporated. Beat in the lemon juice, lemon zest, and vanilla until combined.
Scrape down the sides of your bowl. Beat in the baking powder and salt. Scrape down the sides of your bowl again, and beat for an additional 30 seconds or so. This will ensure the baking powder and salt are well-incorporated into the batter. Note: because we made this a one-bowl recipe and we're skipping combining the dry ingredients beforehand, it's very important to scrape down the sides of your bowl!
Slowly beat in the flour until it's just combined. I like to stop with my mixer when a few streaks still remain and finish mixing with a rubber spatula, just to avoid any over-mixing!
Spoon the batter into your prepared pan and use an offset spatula or knife to spread it into an even layer. Bake the bars for 20 to 30 minutes, or until they're just browned around the edges and the center is set. Tip: a toothpick inserted into the center of the bars should come out with only a few moist crumbs attached to it!
Allow the bars to cool completely in the pan before slicing. If desired, add the remaining 1/2 cup sugar to a shallow bowl and coat the cut lemon cookie bars in sugar before serving.