Preheat the oven to 350°F (180°C) or 160°C if using a fan oven. Line a 9x13-inch (approx 22x23 cm) baking pan with parchment paper and set aside.
In a large bowl, stir together the flour, powdered sugar, granulated sugar and salt.
Pour in the melted butter and the vanilla extract and stir until the dough comes together but is still a bit crumbly.
Transfer the mixture to the prepared baking pan, then pat down into an even layer. Bake in the preheated oven for 15-18 minutes, or until lightly golden.
For the filling:
In a large bowl, whisk together the sugar, flour, lemon zest, and salt. Add the eggs, lemon juice and vanilla and whisk until smooth. You can use an electric mixer to help this along if you like.
Pour the filling onto the warm crust, then return to the oven and bake for an additional 25-30 minutes, or until the filling is completely set and wobbles only slightly.
Remove from the oven, allow to cool, then transfer to the fridge to chill for 3 hours.
Once completely chilled, remove from the baking dish, slice into squares and dust with powdered sugar.
Notes
Press the crust down. In order to get a firm, crispy crust, you want to use your fingers to really press those buttery crumbs down into the pan. Make the crust as even as you can as well so that you don't have some bars with really thick crust and some with very thin crusts.
Use room temperature eggs. Cold eggs are much more difficult to whisk into the other ingredients. Let the eggs sit out for an hour or so before you start to make the lemon curd filling for the bars.
Lightly dust with powdered sugar. The easiest way to do this is to use a fine mesh sieve. Fill it with some powdered sugar and then gently tap it over the bars once they've cooled.
Storing: Keep lemon bars in a sealed container in the refrigerator for up to 5 days, or freeze for up to 3 months.