Mix sugar, sourdough starter, milk, and flour in a bowl. Place the mixture in a warm spot (about 75°F/24°C) for 8-10 hours, or until it has doubled or tripled in size.

Combine flour and milk in a small saucepan. Cook over medium heat, stirring constantly, until the mixture thickens and reaches 150°F/65°C.

Transfer to a bowl and cover with plastic wrap, ensuring the wrap touches the surface to prevent a skin from forming. Let it cool to room temperature.

In the bowl of a stand mixer, combine flour, sugar, eggs, cold milk, all the levain, and tangzhong. Mix until just combined, then let it rest for 30 minutes.

Add salt and knead on medium speed for 5-10 minutes.

Gradually add softened butter, one cube at a time, scraping down the bowl as needed. Continue mixing until the dough passes the window-pane test, which typically takes 15-25 minutes.

Transfer the dough to a lightly oiled bowl and proof for two hours at around 75°F/24°C.

After two hours, place the dough in the fridge to cold-proof for 8-15 hours.

Prepare the filling 30-60 minutes before removing the dough from the fridge.

Melt butter in a small saucepan over low-medium heat. Add dark chocolate and stir until melted. Mix in powdered sugar, cocoa powder, and salt. Stir until smooth.

Transfer the filling to a bowl and chill in the fridge for about 30 minutes, stirring every 5-10 minutes, until it reaches a spreadable consistency.

Remove the dough and filling from the fridge. Stir the filling well.

On a floured surface, roll out the dough into a 13×13 inch (35×35 cm) square.

Spread the filling evenly over the dough, leaving a ½ inch (1 cm) border.

Roll the dough into a tight roulade. Chill seam-side down in the fridge for at least 20 minutes.

Cut the roll in half lengthwise with a floured, sharp knife.

Braid the two halves together, keeping the cut sides facing up.

Tuck the ends in and place the braid into a buttered or lined 10-inch (25 cm) bread tin. Cover and proof at 75°F/24°C for 6-8 hours, or until it fills about 80% of the tin.

Preheat the oven to 390°F/200°C. Bake the babka for 20 minutes, then lower the temperature to 350°F/175°C and bake for another 20 minutes. Cover with foil if it browns too quickly.

While the babka bakes, make the syrup: Heat sugar and water in a small saucepan over medium heat until the sugar dissolves. Let cool.

The babka is done when its internal temperature reaches at least 203°F/95°C and the exterior is golden brown.

Once out of the oven, immediately brush or pour the syrup over the babka. Allow it to cool in the tin before transferring it to a wire rack to finish cooling.