Make the Dough: In the bowl of a stand mixer fitted with a dough hook, use your fingers to rub together the sugar and lemon zest until all of the sugar is moistened. Add the flour and yeast and mix on low speed to combine, about 1 minute. Add the eggs and water and mix on low speed briefly, then increase to medium speed and mix for 5 minutes, until the dough comes together (it will appear very dry and shaggy at first, that's okay).
Add the salt, then add the butter a few cubes at a time, mixing until it is completely incorporated into the dough. Continue mixing on medium speed for about 10 minutes, until the dough is completely smooth, elastic and shiny. Place the dough in a large greased bowl, turning to coat it, then cover with plastic wrap and refrigerate overnight.
Make the Chocolate Filling: Place the chocolate, butter and cocoa powder in a medium bowl and microwave in 30-second increments until completely melted, stirring after each (it will take around 2 minutes). Whisk in the powdered sugar until thoroughly incorporated, then set aside to thicken, about 30 minutes. (The mixture should have the consistency of a soft ganache.)
Assemble the Babka: Divide the dough into two equal pieces. Working with one piece at a time, roll the dough on a lightly floured surface into a 15 x 11-inch rectangle. Use an offset spatula to spread half of the chocolate mixture over the dough, leaving a ½-inch border around the edges. Brush a little bit of water along the long edge of the rectangle farthest away from you. Use both hands to roll up the rectangle, starting from the long side that is closest to you. Press to seal the dampened end, then use both hands to even out the roll. Rest on its seam.
Trim about ¾ inch off both ends of the roll with a serrated knife. Using the same knife, gently cut the roll in half lengthwise, starting at the top and finishing at the seam. With the cut sides facing up, gently alternate crossing the pieces over each other to create a braid. Repeat steps #4 and 5 with the second piece of dough and remaining chocolate filling.
Line a large baking sheet with parchment paper. Gently transfer the two babka braids to the prepared pan and shape them into a circle, braiding together the ends for a continuous effect. Cover with a wet tea towel and allow to rise in a warm place for 1½ hours (the cake will not double in size, but will rise by about 20 percent).
Preheat the oven to 375 degrees F. Remove the tea towel and bake the babka until a skewer inserted into the center of the dough comes out clean and all of the exposed chocolate swirls are firm to the touch, about 25 to 30 minutes.
Make the Simple Syrup: While the babka is baking, make the simple syrup. Combine the water and sugar in a small saucepan and bring to a boil over medium heat. As soon as the sugar dissolves, remove from the heat and set aside.
As soon as the babka comes out of the oven, brush all of the syrup evenly all over the babka (it will seem like a lot, but just keep brushing!). Slide the parchment paper from the baking pan to a wire cooling rack. Allow the babka to cool completely before serving. The babka can be wrapped in plastic wrap and stored at room temperature for up to 3 days, or wrapped in plastic wrap and foil and frozen for up to 1 month.
Notes
If you'd like to bake this chocolate babka as traditional loaves, do the following:
Grease two 9x5-inch loaf pans and line the bottoms with parchment paper. Working with one half of the dough at a time, after slicing the log of dough lengthwise, place the two halves next to each other, cut sides up. Pinch the top ends together, then lift the right half over the left half. Repeat this, but this time lift the left half over the right. Gently pinch together the other ends. Transfer the shaped dough into one of the loaf pans. Repeat with the second half of the dough. Rise as directed above, then bake for 30 to 40 minutes, checking for doneness as directed above. Brush with the simple syrup as directed above, allow to cool for 30 minutes, then remove from the pans and place on a wire rack to cool completely before serving.