In the morning, build a levain: When your starter is mature, mix together 10 grams warm water (80°-85°F), 10 grams all purpose flour, and 10 grams starter. Let rest, covered for 3-4 hours, or until mature, in a warm spot.
In the bowl of a stand mixer, add the flour, milk, 36 grams lightly beaten egg, salt, sugar, and 16 grams of your young mature levain that you built earlier. Use a dough hook and mix on low until incorporated. Let rest, uncovered for 30 minutes.
With the mixer still at medium, add the room temperature butter, a bit at a time, kneading until the butter is completely absorbed before adding the next bit. After all the butter is incorporated, continue to mix until the dough clings to the dough hook and passes the windowpane test. The dough should be smooth and shiny. Shape into a rough ball and transfer to lightly oiled clean bowl, cover, and bulk rise at room temp for 2 hours. The dough will expand but not double. Cover tightly with plastic wrap and place in the fridge overnight.
The next day: Melt the butter and chocolate over a double boiler until smooth then stir in the powdered sugar and cocoa powder until it becomes a spreadable paste. Set aside.
Take the dough out of the fridge and tip out onto a floured work surface. Lightly flour the top and roll out into a long rectangle, about 1/4 inch thick, with the short side closest to your body. Spread the filling evenly on the dough, leaving a 1 inch gap on the edge farthest from you. Starting at the edge closest to you, roll up into a very tight roll, sealing the edge. Place on a lined baking sheet and pop into the freezer for a couple of minutes. Don’t let it sit in there too long because the chocolate will seize up making it difficult to shape.
Line your baking pan with parchment paper, leaving overhang so you can pull your loaf out after it’s baked.
Take the babka out of the freezer, place on a cutting board and slice in half. Pinch two of the ends together, braid, then pinch the ends together. Place in your prepared pan, cover and let proof in a warm spot for about 4-6 hours, or as long as needed – enriched doughs take a particularly long time to proof. The dough should double, rising up to just below the rim of the pan.
Heat the oven to 350°F. When the babka is fully proofed, place the pan on a baking sheet (in case of overflow). Brush with an egg wash (use the remaining eggs from the dough mixed with a bit of water) then bake for 35 - 40 minutes or until golden and the center of the babka reaches 200°F. If the top starts browning too quickly, lower the temperature to 325°F or cover with a piece of foil.
While the babka is baking, make a quick simple syrup by mixing together 1 tablespoon of sugar and 1 tablespoon of boiling water. Stir until the sugar is completely dissolved and let cool.
When the babka is baked through, place the pan on a wire rack. Use an offset spatula to loosen the sides then brush on a thin layer of simple syrup. Let the babka rest for 10 minutes in the pan, then remove immediately by using the parchment overhang. Let rest on a wire rack until cool to the touch, then slice and enjoy!