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To make the dough: In the bowl of a stand mixer fitted with the dough hook, combine all of the dough ingredients. Mix on low speed until combined and no dry spots remain, 1 to 2 minutes. Increase the speed to medium (speed 4 on a KitchenAid) and mix until a smooth dough forms, 6 to 8 minutes; if the dough feels dry or tight, mix in an additional 1 or 2 tablespoons of milk. The dough may not entirely pull away from the sides of the bowl; that’s OK. If mixing by hand, add the dough ingredients to a large bowl and stir to combine. Let the dough rest for 20 minutes, then knead in the bowl until smooth, 2 to 3 minutes.
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Place the dough in a lightly greased bowl, cover, and let rest until puffy and doubled in size, 1 1/2 to 2 hours. Alternatively, proof the dough in the refrigerator for at least 8 hours or up to 1 day; make sure the bowl is covered tightly with plastic or transfer the dough to a large airtight container.
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Turn out the dough onto a lightly floured surface and stretch into a rectangle, roughly 10" x 16". Transfer the dough to a lightly floured baking sheet, cover, and chill in the refrigerator for about 20 minutes to rest.
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To make the filling: While the dough is chilling, in a small bowl, stir together the brown sugar, cinnamon, cocoa, espresso powder, and salt.
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To assemble the buns: Remove the dough from the refrigerator and place on a lightly floured work surface. Flip the dough over and re-flour the surface to ensure the dough doesn't stick. Roll out the dough to a 12" x 24" rectangle, dusting with additional flour as necessary.
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Brush the dough with some of the reserved egg white, applying a thin, even layer. (The remaining egg white will be used to brush the buns just before baking.) Sprinkle the filling mixture over the entire surface of the dough and use your hands to evenly distribute. Sprinkle the chocolate evenly over the filling and gently press to adhere.
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Starting at one of the short sides, roll up the dough into a log and turn seam-side-down. For the cleanest cuts, chill the log in the refrigerator, uncovered, for 10 to 15 minutes before cutting.
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Using a serrated bread knife or sharp chef's knife, cut the log into 12 even slices, roughly 1" to 1 1/4" wide.
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Lightly grease a 9" x 13" pan with nonstick spray and place the buns, cut-side up, in the pan, firmly pressing to flatten to a height of 1", if necessary.
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Cover the pan and let the buns rise in a warm spot until puffy, about 1 1/2 to 2 hours. The buns should completely fill the pan and be touching each other. If your dough was refrigerated in step 2, you might need to add up to 1 hour to this rise time.
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Towards the end of rise, preheat the oven to 400°F with a rack in the middle.
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To bake the buns: Press the center of each bun to again flatten slightly, then brush the buns with the remaining egg white.
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Bake for 18 to 22 minutes, until the tops are deep golden brown.
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To make the glaze: While the buns are baking, in a small saucepan, combine the water, sugar, and salt. Bring to a boil and cook, stirring occasionally, until the mixture thickens slightly and becomes syrupy, about 3 minutes. Remove from the heat and add the orange zest. Cover the saucepan and let steep for 5 to 10 minutes, then strain.
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Remove the chocolate babka buns from the oven and immediately pour the glaze over; brush to evenly distribute. (As the buns cool, any pooled glaze will be absorbed.) Let the buns cool for a few minutes before serving.
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Storage information: Chocolate babka buns are best served the day they’re baked. Store any leftovers in an airtight container at room temperature for up to 3 days. For longer storage, wrap well and freeze for up to 3 months. When ready to serve, thaw at room temperature, wrap in foil, and reheat in a 350°F oven until warmed through.
