Pour warm milk into stand mixer bowl, sprinkle yeast and 1 tablespoon sugar over it, and let sit 5 minutes until foamy. Add remaining sugar, flour, eggs, and salt, then mix on low speed for 2 minutes until shaggy dough forms.
With mixer on medium-low, add softened butter one or two cubes at a time, waiting until each is mostly absorbed. Once all butter is added, knead on medium for 8-10 minutes until smooth, elastic, and slightly tacky.
Transfer dough to a lightly oiled bowl, cover tightly with plastic wrap, and refrigerate for at least 8 hours or up to 24 hours.
Melt chopped dark chocolate and butter in a heatproof bowl over simmering water, stirring until smooth. Remove from heat and stir in sifted powdered sugar, cocoa powder, cinnamon, and salt until a thick spreadable paste forms.
Grease a 9x5 inch loaf pan and line with parchment paper with overhang. On a lightly floured surface, roll chilled dough into a 12x16 inch rectangle.
Spread chocolate filling evenly over the dough rectangle, leaving a 1/2 inch border on the short sides. Roll tightly from the long edge into a 16-inch log and pinch the seam to seal.
Cut the log in half lengthwise, exposing the chocolate layers. Cross the right strand over the left, then left over right, continuing the twist to the bottom. Transfer to the prepared loaf pan.
Cover loosely with plastic wrap and let rise at room temperature for 1.5 to 2 hours until puffed. Near the end, preheat oven to 350°F (175°C).
Bake for 35-45 minutes until deep golden brown and internal temperature reaches 190-200°F (88-93°C). Tent with foil after 25 minutes if browning too quickly.
While baking, combine water and sugar in a small saucepan and heat until sugar dissolves. Brush hot babka generously with the syrup, cool in the pan for 20 minutes, then transfer to a wire rack to cool for at least 30 minutes before slicing.