Make the dough: In the bowl of a stand mixer, mix the yeast with milk and sugar and 1/3 cup of the flour. Cover loosely and let rest until small bubbles appear and break the surface, about 10 to 15 minutes.
Mix in the butter, egg, and salt. Gradually add the flour until it’s incorporated. If using a stand mixer with the dough hook attachment, knead for approximately 5-7 minutes on medium speed until the dough is perfectly smooth and no longer sticky – alternatively, turn out the dough on a lightly floured surface and knead by hand until it is perfectly smooth and no longer sticky, this should take around 15 minutes. Return the dough to the bowl, cover with plastic wrap, and refrigerate for 6 hours.
Prepare the filling: Melt the butter in a medium saucepan. Add the sugar and stir until dissolved or almost completely dissolved. (It’s okay if there are grains of sugar visible – they’ll melt later.) Remove from heat and add the chopped dark chocolate. Let stand 1 minute, then stir until the chocolate is melted and smooth. Stir in the tahini, then the black cocoa powder. Set aside to cool.
Butter a 9-inch loaf pan and line with parchment paper leaving an overhang on the two long sides (which will help you remove the baked babka later).
Assemble the babka: Place the dough on a lightly floured surface and roll out to a 12×16 inch rectangle. If the dough resists and contracts when rolling, let it rest for 5-10 minutes and then continue rolling. Spread the filling in an even layer leaving about a 1-inch border. (Don’t worry if the edges are a bit uneven – no one will be able to tell once the babka is baked!).
Starting at one of the long sides, roll up dough so that the seam is facing downwards. With a sharp knife, cut the roll in two halves along the long side. Turn the dough slightly to ensure the cut sides are facing upwards on both pieces. Pinch the ends together on one side then gently twist the two pieces of dough, pinching the other ends together as well. Carefully push on both ends to compress the twisted strands of dough until they are about the length of your loaf pan. Using both hands, carefully place the babka into the prepared loaf pan. Cover and let proof somewhere warm for about 60 – 90 minutes, or until risen and puffy. Preheat oven to 375°F.
Make the simple syrup: While the dough is rising, bring the water, sugar, honey, and cardamom to a boil in a small saucepan. Let boil for 4 minutes, skimming off any foam that rises to the surface with a spoon. Remove from heat and set aside.
Bake the babka on the middle rack of the oven for 30 minutes, or until a skewer inserted into the center (in a part where there is less chocolate filling), comes out clean of dough. There may be some chocolate clinging to it, which is normal.
Immediately spoon the room temperature syrup over the babka (you may not need all the syrup) and let cool completely before lifting the babka out.
The babka should keep 3-4 days at room temperature.