Trace a 6 1/2-inch circle onto a large, flexible, plastic lid, such as one from a coffee can. Using scissors, cut lip from lid. With utility knife, cut out circle shape to make stencil.
Heat oven to 400 degrees F.
In bowl of electric mixer fitted with paddle attachment, beat butter and sugar on medium speed until fluffy, about 5 minutes. Beat in egg whites, one at a time. Add flour and beat to combine.
Line baking sheet with silpat (French nonstick baking mat) or parchment paper. Place stencil on mat. Using an offset spatula, spread a thin layer of batter over stencil. Gently remove stencil. Repeat, making 1 more circle. Bake until golden around the edges, about 4 minutes. Working quickly while tuiles are still warm, use a spatula to remove from baking sheet and drape each cookie over a coffee cup. Cover with a clean kitchen towel and gently press to help form cup shape. If cookies cool before shaping, return pan to oven for about 20 seconds. Spread and bake remaining batter. Store in an airtight container up to 1 week.
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