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Preheat oven to 325°F. Coat a heavy baking sheet well with cooking spray or line with parchment paper.
In a bowl whisk together ground almonds, potato starch, sugar, and salt and whisk in whites, butter, zest, and vanilla extract until combined well. Drop rounded teaspoons batter about 4 inches apart onto baking sheet and with back of a spoon spread into 3 ½-inch rounds. Try to keep these distances - if you' let your cookies get too close, they'll morph into one giant cookie slab. This is fine too, but it won't look as nice when cut into squares. Sprinkle each cookie with about ½ tablespoon sliced almonds and bake in middle of oven 8 minutes, or until golden. If your oven cooks unevenly, take the pan out and turn it halfway through.
Working quickly, remove cookies, 1 at a time, from baking sheet with a thin spatula and rest on a cooling rack (the original recipe recommends curling these. I usually pass on this detail, as it makes the cookies harder to stack). Cool cookies completely and transfer to an airtight container.
Make more cookies with remaining batter in same manner, spraying or lining baking sheet for each batch. Tuiles may be made 2 days ahead and kept in an airtight container at room temperature.