Warm milk in the microwave for 1 minute on high heat. In a large mixing bowl of a stand mixer, mix together the yeast, milk, and sugar (or if you are making by hand, use a large mixing bowl). Allow it to sit for about 5-10 minutes until the yeast is bubbly.
Add the salt, eggs, egg yolk, vanilla, and flour (1/2 cup at a time) and knead until the dough pulls away from the sides of the bowl and is tacky and not too sticky.
While kneading, add the softened butter 1 tablespoon at a time until it is fully incorporated into the dough. The dough should be fairly sticky.
Cover the dough and allow it to rise 1 1/2-2 hours until the dough has doubled in size. Push the dough down, cover and chill in the refrigerator for 45 minutes.
While the dough is chilling, make the filling by beating together the butter, powdered sugar, cocoa powder, and cinnamon. If the filling is too thick, thin it down by adding milk, as needed, to make a spreadable paste.
Remove the dough from the bowl onto a lightly floured surface and roll into a large rectangle about 20 inches long. Spread the chocolate filling on the dough leaving about a 1 inch border around the edges. Sprinkle the chopped chocolate bar on top of the filling.
Roll the dough lengthwise like a cinnamon roll and divide the dough in half, creating 2 approximately 10" tubes. Cover the dough tubes with saran wrap and freeze for about 10 minutes.
Cut each tube lengthwise to expose the chocolate filling. Twist each piece about 3-4 times lengthwise then twist 2 pieces together to form the babka and place in a greased loaf pan. Cover and allow the dough to rise again for about 45 minutes to 1 hour or until it has doubled in size.
Create an aluminum foil tent over each loaf pan and bake at 350 degrees F for 20 minutes. Remove the foil and continue baking for an additional 10-20 minutes or until the internal temperature is 185 degrees F. Watch closely as it will burn easily.
While it is baking, create the sugar syrup by mixing together the sugar and water in a pot over the stove. Bring the mixture to a boil, whisking frequently. Allow it to boil for about 3-5 minutes, continuing to stir frequently. Remove from the heat and set aside.
Brush the syrup onto the baked loaves and allow them to cool, in the pan, for 30 minutes. Remove them from the pan onto a wire rack to cool fully. Slice and enjoy!