Make The Dough. In a mixing bowl, with an electric mixer, stir together flour, sugar, and salt together in the bowl. Add starter, eggs, yolk, milk, and vanilla using the dough hook. Mix on low speed until mixture comes together, scraping down the bowl and hook as needed. On low speed, add butter, 1 tablespoon at a time, until incorporated. Let the dough rest for 20 minutes.
Knead dough. Increase speed to medium and knead with the dough hook for about 10 minutes. The dough should have pulled away from the sides of the bowl.
Bulk Rise. Place the dough in an oiled bowl, turning the dough, so it is evenly coated. Cover with a damp towel and let rise for 6-8 hours at room temperature (75-78°F). I don't recommend doing this in the fridge as it will be too cold and not get a sufficient rise.
Prepare Pans. Grease, or butter two 8 ½” loaf pans. Line pans with parchment paper.
Make filling. Place butter, sugar, cocoa, and cinnamon in a mixing bowl and beat until combined. It should be a very soft paste that you will be able to spread easily.
Shape Babka. Divide dough in half. On a floured surface, roll one dough half into a rectangle about 9 x 12 inches. Spread on half of the filling. Sprinkle with half of the chopped chocolate and press down lightly. Roll up the long side of the dough fairly tightly into a log. Using a sharp knife, cut the log in half, lengthwise down the center. Twist the two cut halves together, by alternating each half over each other, pinching the ends together, keeping the middle exposed. Place in one of the prepared pans. Repeat with the second dough.
Proof. Cover the pans with greased plastic wrap and let rise at room temperature for 2-4 hours until puffy. During the last 20 minutes of rising, preheat the oven to 350ºF.
Make the syrup. Combine the sugar and water in a pan. Bring to a boil over medium heat, stirring frequently. Let the mixture boil for a minute or two, then remove from heat to cool.
Bake. Bake the breads in the oven for 40–50 minutes, or until 190ºF. Place on a wire rack and poke the breads several times with a long skewer.
Add syrup. Pour the cooled syrup over the babkas ¼ cup at a time, dividing the syrup evenly. Let the babkas cool in the pans for about 40 minutes, then use the parchment to lift out of the pans. Let cool completely on a wire rack.