Using a food processor, gently pulse the flour, salt and sugar to mix. Then cut the butter in chunks, add to the flour mixture and pulse so you get fine crumbs. Add the egg and vanilla and pulse again a few times so the mixture starts coming together as coarse crumbs, in small clumps. Remove the blade and press the mixture together with your hands. Alternatively, you can mix by hand by rubbing the butter into the flour-sugar-salt mixture to form coarse crumbs. Separately, gently beat the egg before adding it and the vanilla. Mix in gently and bring together.
However you mix the dough, after you bring the dough together, flatten it into a disc, wrap and refrigerate for at least 30 minutes, but 45 minutes is probably better, so that it firms up before you roll it out.
Line two baking sheets/trays with parchment paper or silicone baking mats. When well chilled, cut the dough in two equal pieces and re-wrap and chill the one you are not working with. Roll the one out into a square approximately 8 - 8.5 inches square (20 - 21.5cm). If the dough doesn't roll out quite perfectly, you can trim wider edges and stick them elsewhere as you are getting towards the final size and they should stick fine when pressed on and rolled further (though try not to do too much patching especially once it's about the final size). Carefully transfer to one of the lined baking sheets and chill. Then, roll the other piece to the same size and transfer to the other baking sheet. The exact size is not too important, it just gets a little more delicate if thinner, but the main thing is both pieces should be the same size.
Preheat oven to 350F/180C. As the oven warms up, chill the dough squares further so they are as cool as possible when baked. When the oven comes to temperature, place both baking sheets in oven.
Bake the squares for approximately 10 - 15 minutes until just starting to become golden brown in color around the edges - they will still be slightly soft to touch but will look dry.
Let them cool for around 10 minutes so that they are firmer but still slightly warm.