Combine flour and sugar in the bowl of a stand mixer fitted with the dough hook attachment.
Once combined, add the sourdough starter, eggs, milk, and vanilla using the dough hook. Mix on low speed until the mixture comes together. Continue on low speed, and add the butter 1 tbs at a time, until incorporated. Let the dough rest for 15 minutes.
After 15 minutes, add your salt then continue kneading on low-medium speed for another 15 minutes until the dough passes the windowpane test.
For the bulk rise, place the dough in a greased bowl covered with a damp towel or beeswax wrap for about 6-8 hours at room temperature.
After the bulk rise, if you have time in the day you can bake the babka, or you can place the dough in the fridge until the morning.
If you placed the babka in the fridge overnight, remove about 2-4 hours before you want to bake so it can come back to room temperature.
Grease 2 loaf pans and make the filling by combining the softened butter, honey, cocoa powder, and chocolate chips.
On a floured surface, divide the dough in half then roll one dough half into a rectangle about 9 x 12 inches and spread half the filling leaving about an inch of space on the perimeter of the rectangle.
Roll up the long side of the dough tightly into a log. Pinch the ends and seem of the babka log so it creates a smooth log.
If you're pinched for time continue to the next step, if not, place the log into the freezer for 15 minutes.
Then using a sharp, greased knife, cut the log in half, lengthwise down the center. Braid the two halves together, by alternating each half over each other, then pinch the ends together. Place in one of the prepared pans.
Repeat with the second dough.
Pour 1/2 cup of heavy whipping cream over each babka and allow to rise covered in a warm spot for about 1-4 hours until puffy.
During the last 10 minutes of rising, preheat the oven to 350ºF.
Bake the sourdough babka in the oven for about 40-45 minutes. When finished baking, let them cool in the pan for about 20 minutes then serve.