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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess.
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To make the dough: In the bowl of a stand mixer fitted with the dough hook, combine all the dough ingredients. Mix on medium-low speed until a shaggy dough forms, about 1 minute.
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Scrape the bottom and sides of the bowl. Increase the speed to medium-high, and continue mixing until a smooth, tacky dough forms, about 2 minutes, stopping to scrape the bowl once halfway through. At the end of mixing, the dough will have begun to clear the sides of the bowl and should start to gather around the dough hook.
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Cover the dough and let it rise at room temperature overnight (at least 12 hours), or up to 24 hours, based on your schedule. It should be very puffy and more than double in size.
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To make the filling: In a small bowl, mix the sugar, cocoa, cinnamon, and salt until well combined. Add the melted butter and stir to form a thick, granular paste.
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To shape the babka: Lightly grease a 9" x 4" half Pullman loaf pan or 9" x 5" loaf pan. Line with a parchment sling and lightly grease the parchment. Use a bowl scraper to ease the dough out of the bowl onto a lightly floured work surface. Gently press the dough to deflate it and sprinkle with additional flour.
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Use a floured rolling pin to roll the dough to a rough 16" x 12" rectangle of even thickness with one short side facing you. If the dough resists rolling or snaps back, cover and let it rest for 10 to 15 minutes, then try again.
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Once the dough is rolled out, dollop the filling mixture on top in small mounds. Use a small offset spatula to spread it into a thin, even layer, leaving a 1" border of dough bare along the short side farthest away from you. Sprinkle the chopped chocolate evenly over the filling mixture.
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Beginning with the short side closest to you, roll the dough into a log. Pinch the seam shut to seal the filling inside.
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Place the log perpendicularly in front of you with the seam-side facing up. Use a bench knife or sharp serrated knife to cut the log in half lengthwise along the seam.
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With the cut sides facing up, make an “X” with the two pieces of dough. Twist the two pieces together, then pinch the ends to seal. Gently tuck the ends under the loaf to hide the pinched area.
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Transfer the dough to the prepared pan, then cover and let rise at room temperature until the crest of the loaf is about 3/4" to 1" below the rim of the pan, about 2 to 4 hours depending on the temperature of your kitchen. Toward the end of the rising time, arrange a rack in the lower third of the oven and preheat it to 350°F.
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When ready to bake, brush the top of the loaf with the egg wash. Bake the sourdough babka until the top crust is golden brown, and the internal temperature reaches at least 200°F when measured with a digital thermometer, 45 to 55 minutes. If the top is browning too quickly, tent the loaf with foil.
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To make the glaze: While the babka is baking, combine the glaze ingredients in a small saucepan. Bring the mixture to a boil over medium heat while stirring. Reduce the heat to maintain a simmer for 5 to 7 minutes, or until the sugar has dissolved and the mixture has thickened slightly. Set aside until ready to use.
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Remove the babka from the oven, then brush the syrup over the top of the loaf until all the syrup is used. Let the loaf cool in the pan for 10 minutes, then remove from the pan and transfer to a wire rack to cool completely before slicing and serving.
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Storage information: Store leftover sourdough babka in an airtight container at room temperature for up to 3 days or freeze for longer storage.
