Preheat the oven to 350°F. Grease a 9x13 inch baking dish or line the baking dish with parchment paper, to make it easier to remove the lemon bars.
In a large mixing bowl, cream together the softened butter, granulated sugar, and vanilla extract until the mixture is light and fluffy.
Gradually add the flour and salt to the butter mixture, mixing until the ingredients are crumbly to make a shortbread mixture. Press the shortbread mixture evenly into the bottom of the prepared baking dish to form the crust.
Bake the crust in the preheated oven for 15 to 20 minutes, or until the crust is lightly golden brown. Let the crust cool for a few minutes before adding the lemon filling.
While the shortbread crust is baking, prepare the lemon filling. Add 2 cups of granulated sugar and ½ cup of flour to a large mixing bowl. Whisk the flour and sugar together until they are well blended.
Add the eggs, egg yolks (optional), lemon juice, lemon extract (optional), salt, and lemon zest to the bowl, and whisk until the ingredients are well combined and smooth. Cover and refrigerate the lemon filling until the shortbread crust is baked and cooled.
Once the shortbread crust is baked, remove it from the oven and pour the lemon filling over the baked crust, spreading it evenly.
Return the baking dish to the oven and bake for an additional 20 to 25 minutes, or until the filling is set and the edges are lightly golden brown.
Allow the lemon bars to cool completely in the baking dish on a wire rack. Once cooled, refrigerate the lemon bars for at least 1 hour to firm up the filling before cutting the dessert into squares or bars.
If desired, dust the top of the cooled lemon bars with powdered sugar before serving.