Beat butter and sugar together with an electric mixer on medium-high speed. Beat in the egg whites, one at a time. Beat in the vanilla.
Lower the speed and add the flour or flour-cocoa mixture (see Cook's Note) and mix until just combined. Do not overmix. Cover bowl and chill for at least 1 hour.
Preheat oven to 325 degrees F (165 degrees C).
Cut a small hole (about 3 1/2 inch diameter) in a thin piece of cardboard or plastic to serve as a template in forming the tuiles. Line a cookie sheet with parchment paper or a silicon mat and place the stencil on it. Using a small off-set spatula, place a small amount of the batter in the center of the hole of the stencil and spread it out evenly. Carefully lift the stencil off. Repeat for more cookies.
Bake in preheated oven until lightly brown around the edges, 8 to 10 minutes. Remove from the oven and while still hot, remove tuiles from the pan and drape them over a rolling pin. Allow to sit a few minutes to harden and cool. Store in an airtight container.