Combine sugar, flour, baking powder, and salt in a medium mixing bowl. Mix until evenly combined.
Add in vanilla extract and melted butter and roughly mix with a spoon.
Add in one egg white and mix the dough until it starts to come together. Knead gently with your hands until all of the dry spots have been hydrated.
We like to chill the dough so it's easier to work with, but you can try to roll this out right away if you would like! Otherwise, wrap the dough in plastic wrap and chill the dough for 15-30 minutes.
While the dough is chilling, preheat oven to 365F (185C).
Flour a work surface and roll out the dough, adding flour to the top as well so it doesn't stick to the rolling pin. Roll the cookies out to between 1/8" and 1/4" (the thickness is up to you - the thinner they are, the crispier they will be!)
Cut into circles about 2" in diameter (we just use a glass to do this step). Transfer the cookies to a parchment or silicone mat lined baking sheet.
Bake for about 7-8 minutes, until the edges turn golden. Transfer to a cooling rack.
Collect any dough scraps and roll the dough out again on your floured surface; we like to do this while the first batch of cookies are baking, but you can also bake two trays at a time! Continue to use the scraps until you are out of dough, or the dough is too dry to hold together any longer; and continue baking the cookies for 7-8 minutes at a time.
Enjoy on their own with a cup of tea, or use for sandwich cookies, or even as a layer in a trifle!