Make the Dough:

1. In the bowl of a stand mixer, combine warm milk, 1 tablespoon of the sugar, and yeast. Let stand for 5-10 minutes until foamy and activated.
2. Add remaining sugar, flour, salt, eggs, and vanilla to the yeast mixture. Using the dough hook attachment, mix on low speed until ingredients come together, about 2 minutes.
3. Increase speed to medium-low and knead for 5-6 minutes until the dough becomes smooth. Add butter pieces gradually, a few at a time, allowing each addition to incorporate before adding more. Continue kneading for another 5-7 minutes until the dough is elastic, smooth, and slightly tacky but not sticky.
4. Transfer dough to a lightly greased bowl, cover with plastic wrap or a damp kitchen towel, and let rise in a warm place for 1.5-2 hours until doubled in size.

Prepare the Chocolate Filling:

5. While the dough rises, make the filling. Place chopped chocolate in a heatproof bowl and microwave in 30-second intervals, stirring between each, until melted and smooth. Let cool slightly.
6. In a medium bowl, whisk together melted chocolate, melted butter, cocoa powder, sugar, vanilla, and salt until completely smooth and spreadable. Set aside at room temperature.

Shape the Babka:

7. Once dough has doubled, punch it down and turn it out onto a lightly floured surface. Divide dough in half. Work with one half at a time, keeping the other covered.
8. Roll one portion into a large rectangle, approximately 10×16 inches. The dough should be about ¼-inch thick.
9. Using an offset spatula, spread half of the chocolate filling evenly over the dough, leaving a ½-inch border on one long side.
10. Starting from the long side with filling, tightly roll the dough into a log, like a jelly roll. Pinch the seam to seal.
11. Using a sharp knife, cut the log lengthwise down the middle to create two long halves, leaving about 1 inch intact at one end to keep them attached. Turn the cut sides up to expose the layers.
12. Twist the two halves around each other, keeping the cut sides facing up as much as possible to show off those beautiful chocolate swirls. Tuck the ends under.
13. Carefully transfer the twisted dough to a greased 9×5-inch loaf pan. Cover loosely and let rise for 1-1.5 hours until puffy and nearly doubled.
14. Repeat steps 8-13 with the second portion of dough and remaining filling.

Bake the Babka:

15. Preheat oven to 350°F (175°C). Position rack in the center of the oven.
16. Make egg wash by whisking together egg and water. Brush generously over the risen babka loaves, making sure to get into all the nooks and crannies.
17. Bake for 40-45 minutes until deeply golden brown and an instant-read thermometer inserted into the center reads 190°F. If the tops are browning too quickly after 30 minutes, tent loosely with aluminum foil.

Make Simple Syrup and Finish:

18. While babka bakes, make simple syrup. Combine sugar and water in a small saucepan over medium heat. Stir until sugar dissolves completely, then simmer for 2-3 minutes. Remove from heat.
19. As soon as the babka comes out of the oven, generously brush the hot loaves with simple syrup while they’re still in the pans. Let them soak in for 5 minutes.
20. Carefully remove babka from pans and transfer to a wire rack. Brush any remaining syrup all over the loaves. Let cool for at least 30 minutes before slicing (if you can wait that long!).