Line a heavy baking sheet (or an insulated sheet or two sheets sandwiched together) with parchment or butter it. Roll the dough into a rectangle about 13x16 inches and about 1/8 inch thick. Stir the chilled pastry cream to soften it and then spread it over the dough in a thin layer. Scatter the chocolate chips evenly over the surface. Roll the rectangle into a cylinder from the wider side and pinch the long edge to seal. Put the cylinder on the baking sheet. Cut the cylinder in half lengthwise, splitting it into two thin strips. Arrange the strips parallel to one another so that the filling is facing up, push them together, and wrap them around each other to form a twist, working from the center.

Position a rack in the middle on the oven and heat the oven to 350 degrees F. Cover the shaped dough and proof until it's large, puffy, and remains indented when lightly pressed with your fingertip, about 45 minutes.

Brush the braid with the beaten egg, taking care not to smear the filling or dislodge the chocolate bits. Bake until golden brown, about 35 minutes, rotating the pan halfway through baking. Let cool on a rack for 1 hour before slicing.

In a large bowl, mix the 3 tablespoons flour with the yeast and then whisk in the water. Let the mixture sit covered until it has begun to puff, 10 to 15 minutes.

Fit a large-capacity food processor with the metal blade. Put the remaining flour in the workbowl and then add the yeast mixture, potato, egg yolks, vanilla, and sour cream. Process the dough for about 1 minute. Remove it from the machine and knead it by hand on an unfloured countertop for 1 minute to redistribute the heat. The dough will be very stiff at this point. Continue this alternating kneading: process for 30 seconds and then knead on the counter for about 30 seconds, until the dough is very smooth (this should take 2 to 3 processing rounds).

Put the dough back in the food processor and add the sugar and salt, kneading again in the processor and then on the counter until the sugar has dissolved (the dough will soften considerably and become very sticky; this is fine).

Finally, return the dough to the processor, add the butter, and do another alternating kneading round until the butter is well incorporated and the dough is very soft and smooth, about 1 minute. The dough won't clean the bowl at this point. It's all right if it feels quite soft and warm after processing: kneading the dough on the counter will help it cool down and firm up.

Transfer the dough to a container at least four times its volume (no need to grease the container); seal well. (At this point, the dough can instead be rolled in flour and then sealed in a plastic bag and refrigerated for up to 4 days. If you do mix ahead and chill the dough, pull it out of the fridge 3 to 4 hours before baking.) Let the dough ferment at room temperature for about 3 hours or until it's expanded to 3 times its volume and an indent remains when you press it with a floured finger.

In a medium saucepan, warm the milk over medium heat (if you're using a vanilla bean, add it now) just until a skin forms. Take the pan off the heat. In a medium mixing bowl, combine the sugar, flour, and salt. Add the yolks, beating with a wooden spoon. Whisk in the warm milk in a thin stream, whisking constantly. Return the milk mixture to the saucepan. Cook over medium heat, whisking constantly, until the mixture is extremely thick and gluey (you'll need to switch to a wooden spoon), about 5 minutes. If you're using vanilla extract, stir it in now. Immediately force the pastry cream through a sieve. Gently press a sheet of waxed paper or plastic onto the surface of the hot pastry cream to prevent a skin from forming. Let cool and then refrigerate until ready to use it.