Instructions

for the dough

 Whisk flour, yeast, and salt together in bowl of stand mixer. Whisk water, eggs and yolk, and sugar together in 4-cup liquid measuring cup. Using dough hook on low speed, slowly add water mixture to flour mixture and mix until cohesive dough starts to form and no dry flour remains, about 2 minutes, scraping down bowl as needed.
 Increase speed to medium-low; add butter, 1 piece at a time; and knead until butter is fully incorporated, about 4 minutes. Continue to knead until dough is smooth and elastic and clears sides of bowl, 11 to 13 minutes.
 Transfer dough to lightly floured counter and divide in half. Knead half of dough by hand to form smooth, round ball, about 30 seconds. Place dough seam side down in lightly greased large bowl or container. Return remaining half of dough to mixer, add cocoa, and knead on medium-low speed until cocoa is evenly incorporated, about 2 minutes. Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 30 seconds. Place second dough ball seam side down in second lightly greased large bowl or container. Cover bowls tightly with plastic wrap and let rise at room temperature until increased in size by half, 45 minutes to 1 hour.

for the filling

 Microwave chocolate, butter, and cocoa in bowl at 50 percent power, stirring occasionally, until melted and smooth, about 2 minutes. Stir in sugar until combined and let cool completely, about 30 minutes. Whisk in egg white until fully combined and mixture turns glossy.
 Line baking sheet with parchment paper and lightly flour. Transfer plain dough seam side down to prepared sheet and press into 6 by 6-inch square. Spread filling over dough, leaving ¼-inch border around edges. Press chocolate dough into 6 by 6-inch square on lightly floured counter. Place on top of filling and press gently to adhere. Cover loosely with greased plastic and refrigerate for 30 minutes.
 Line 2 rimmed baking sheets with parchment paper. Transfer chilled dough square to lightly floured counter. Press and roll dough into 12 by 16-inch rectangle with short side parallel to counter edge. Using sharp pizza wheel or knife, starting at 1 short side, cut dough into twelve 16 by 1-inch strips. Working with 1 dough strip at a time, tightly coil ends of strip in opposite directions to form tight S shape. Arrange buns on prepared sheets, six per sheet, spaced about 2½ inches apart. Cover loosely with greased plastic and refrigerate for at least 2 hours or up to 24 hours.
 One hour before baking, remove buns from refrigerator and let sit at room temperature. Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Gently brush buns with egg mixture and bake until golden brown, 18 to 22 minutes, switching and rotating sheets halfway through baking. Transfer buns to wire rack and let cool completely, about 30 minutes. Serve.