In a mixing bowl, combine your unfed sourdough starter, all-purpose flour, room temperature water, and light brown sugar. Stir until well incorporated. Cover and let it ferment for 6-8 hours.

In the bowl of a stand mixer, combine the prepared sweet starter, additional flour, cornstarch, large eggs, whole milk, salt, and granulated sugar. Mix on low speed, then gradually add softened unsalted butter. Knead for 8-10 minutes.

Transfer the kneaded dough to a greased bowl, cover, and let it rise for 4-6 hours or overnight in the fridge until nearly doubled in size.

Warm the heavy cream in a saucepan, then pour over chopped dark chocolate and let sit. Stir in cocoa powder, sugar, softened butter, vanilla extract, and walnuts. Cool to room temperature.

Roll the risen dough into a rectangle, spread the chocolate filling, roll it into a log, slice in half lengthwise, and twist into a loaf pan.

Cover the loaf pan loosely and let it rise for 2-4 hours until puffy.

Preheat oven to 350°F (175°C). Bake for 35-40 minutes until golden brown. If browning too quickly, cover with foil.

Heat water and granulated sugar in a saucepan to make syrup. Brush over the warm babka after baking. Cool in the pan for 10-15 minutes before transferring to a wire rack.