Preheat oven to 475 degrees.
In the bowl of an electric mixer fitted with the paddle attachment, beat butter and sugar on medium until well combined, about 2 minutes. Add flour and cocoa and mix to combine. Gradually beat in egg whites and continue to beat for 4 minutes.
Trace a 5-inch circle onto a large, flexible plastic lid or cutting mat. Using scissors, cut lip from lid. With a utility knife, cut out circle to make a stencil.
Place stencil on a baking sheet. Using a small offset spatula, spread a thin layer of batter over stencil; carefully lift up stencil. Transfer to oven and bake for 2 to 3 minutes. Using a spatula, roll circles around a wooden reamer to form a cone; let cool completely. Repeat process to make 16 cones.