Preheat the oven to 350°F. Line as many as 3 sheet pans with silicone baking liners or brush nonstick pans with room-temperature butter.
In a bowl, combine the flour, sugar, and orange zest. Add the egg whites and stir with a whisk until smooth. Add the vanilla and butter and stir again until smooth. Gently stir in the almonds.
Measure out a mounded tablespoon of batter onto a prepared baking sheet. With the back of a spoon or a small offset spatula, spread the batter - if you're using a spoon, use a circular motion - until it forms a thin 4-inch-diameter circle. Repeat until you have at least 6 circles - 2 rows of 3 each - on each of the sheet pans.
Stagger the baking so that no more than 1 sheet pan comes out of the oven at a time. Bake for 13 to 15 minutes, until the edges of the tuiles are deep brown and the insides are pale brown.
Working very quickly, lift the still-flexible tuiles off the sheet pan with a metal spatula and immediately place each one over a rolling pin, top side up, or place them in a tuile mold. If they don't droop on their own, press them gently against the pin. Store tightly sealed for up to 1 week.