Preheat your oven to 350°F (175°C). In a medium bowl, whisk together the flour, powdered sugar, and salt. Add the cold, cubed butter. Use a pastry blender or your fingers to cut the butter into the flour mixture until it resembles coarse crumbs. This step is essential for creating a tender crust.
Press the shortbread mixture evenly into the bottom of an ungreased 9x13 inch baking pan. You can use the bottom of a measuring cup to help flatten and even out the surface. Bake the crust for 18-20 minutes, or until it's lightly golden brown. While the crust is baking, you can prepare the lemon filling.
In a large bowl, whisk together the granulated sugar, flour, baking powder, and salt. Add the eggs one at a time, whisking well after each addition. Stir in the lemon juice and lemon zest until everything is well combined and smooth.
Once the crust is lightly golden, remove it from the oven. Pour the lemon filling evenly over the hot crust. Return the pan to the oven and bake for another 25-30 minutes, or until the filling is set and no longer jiggles. The center might still have a slight wobble, but it will firm up as it cools.
Let the Lemon Bars with Shortbread Crust cool completely in the pan on a wire rack. Once cooled, refrigerate for at least 2 hours before cutting into squares. This will make them easier to cut and prevents the filling from being too soft. Dust with powdered sugar before serving, if desired.