Preheat an oven to 375 degrees F (190 degrees C). Lightly grease 2 baking sheets.
Beat the butter, sugar, and vanilla with an electric mixer in a large bowl until smooth. Add the egg whites one at a time, allowing each to blend into the butter mixture before adding the next. Mix in the flour until just incorporated.
Drop teaspoonfuls of the dough onto ungreased baking sheets. Spread thinly with a spatula or the back of the spoon into a 3 inch circle.
Bake in the preheated oven until browned along the edges, about 5 minutes.
Remove the baked wafers from the baking sheet one at a time using a spatula or kitchen turner. Roll each wafer around the handle of a wooden spoon until the edges overlap. Cool seam-side down on a wire rack until completely crisp.