Preheat the oven to 350 degrees. Line a 9x13 inch baking dish with parchment paper.
To make the shortbread crust, in a large bowl, whisk together the 2 1/2 cups of flour, powdered sugar and salt. Add the melted butter and vanilla and stir with a spatula until blended (the mixture will be crumbly). Transfer the dough to the prepared pan and press into a flat, even layer.
Bake for 15 to 20 minutes, until light golden.
Meanwhile, to make the filling, in a large bowl, whisk together the eggs and granulated sugar until blended. Add the lemon juice and lemon zest and whisk to combine. Whisk in the 1 cup of flour and whisk until smooth and blended, with no traces of flour.
Pour the filling over the warm crust. Bake for 25 to 35 minutes, until the filling is almost set (to prevent over-baking, remove the pan from the oven as soon as the filling is slightly jiggly in the center).
Cool in the pan, on a wire rack, at room temperature for 1 hour, then refrigerate for at least 2 hours.
When ready to serve, use the parchment paper to lift the bars from the pan and transfer them to a cutting board. Dust with powdered sugar and cut into 18 to 24 bars using a clean, sharp, damp knife. Clean and re-wet the knife after each cut.
Store leftovers in an airtight container in the refrigerator for up to 4 days or freeze for up to 3 months. There’s no need to thaw frozen lemon bars. Dust with more powdered sugar before serving if desired.