In a mixing bowl, stir together the beaten egg white with the sugar, both flours, and vanilla.
Pour the melted butter slowly into the mixture and combine well. Chill for at least 4 hours.
Preheat oven to 350 degrees. Grease a cookie sheet, line with parchment paper, and spray with nonstick spray (or use a silicone baking mat).
Using a spoon, put dollops of batter on the prepared surface, spreading each one into a disk 1½ to 2 inches in diameter.
Bake until very light brown around the edges, about 10 minutes.
To make decorative forms, set out some clean wooden spoons, a rolling pin, or even a wine bottle.
When the tuiles are cooked but still soft, carefully lift them with your fingers or a spatula and place each one (smooth side up) on the handle of a wooden spoon or other curved surface, letting it harden into a pretty shape.
If the tuiles in the pan start to harden, put them back in a warm oven for a minute. (These cookies are just as tasty left flat, and you can also bake the batter in one large sheet and break it into small, irregular pieces.)