Feed your sourdough starter the evening before so it is very bubbly and active by morning.

Combine all the dough ingredients in the bowl of a stand mixer fitted with a dough hook. Knead until smooth and glossy, 10 to 15 minutes. The dough will be very wet; do not add more flour.

Cover the bowl and let the dough bulk ferment in a warm spot until doubled, usually 6 to 8 hours. Watch the dough and pull it at a true double rather than by the clock, since a long over-ferment can leave the center gummy.

Place the dough in the refrigerator, ideally overnight or for at least 3 to 4 hours, so it firms up well for shaping. A thorough chill makes this wet dough much easier to roll and braid.

Make the filling: warm the cocoa powder, sugar, chocolate chips, and softened butter in a small pot over low heat, stirring, until melted and smooth. Let cool slightly.

Line a 9 by 5 inch loaf pan with parchment, leaving an overhang on the long sides. Butter the pan first so the parchment stays put.

On a lightly floured surface, roll the chilled dough into a rectangle about 10 by 14 inches. Spread the filling to the edges, leaving 1 inch bare along one short side. Sprinkle with the chopped chocolate.

Roll the dough up into a tight log from the short side and pinch the seam to seal. Cut the log in half lengthwise to expose the layers, then twist the halves together, cut sides up, into a braid.

Settle the braid into the prepared loaf pan, tucking the ends under if needed. Loosely cover and let rise in a warm spot for about 4 hours, until puffy and risen to the top of the pan.

Preheat the oven to 350°F. Whisk 1 egg with a splash of water and brush over the top. Bake on a lower-middle rack for about 40 to 50 minutes. Tent loosely with foil as soon as the top is golden (around 20 minutes) so it does not over-darken, and drop the oven to 325°F if it is still browning too fast. Go by temperature: the center is done at 190 to 195°F in the thickest part, which finishes last, so give it another 5 minutes under the foil if the middle is still soft. Cool completely, at least a couple of hours, before slicing.

Cool in the pan for about 15 minutes, then lift out by the parchment onto a rack. If using, simmer the simple syrup until dissolved and brush it over the warm babka. Cool completely before slicing, or the center may seem gooey.