In a mixing bowl, cream butter with 1/2 cup sugar on low speed until light yellow.
Gradually mix in all-purpose flour, vanilla extract, and a pinch of salt until crumbly but binding.
Press the mixture firmly into a greased 9×13-inch baking pan using your palms to ensure it holds together like putty.
Bake the crust at 325°F for 12 minutes until slightly golden around the edges.
Chill the crust in the refrigerator for 20 minutes to firm it up.
While the crust chills, zest the lemons into a large mixing bowl and set aside.
Whisk the eggs and gradually add 1 1/4 cups granulated sugar while whisking until fully dissolved.
Slowly add the freshly squeezed lemon juice (about 1/2 cup total from 2 large lemons) to the mixture, continuing to whisk constantly.
Pour the lemon filling over the chilled crust and smooth the top with an offset spatula.
Bake at 350°F for 30–35 minutes until the middle is cooked through and the edges are golden.
Cool the pan completely at room temperature, then chill in the refrigerator for a minimum of 2 hours before cutting.
Use a sharp knife to slice into 16 bars.