To prepare the Chocolate Babka, begin by placing the warm milk in the bowl of a stand mixer equipped with the whisk attachment. Sprinkle the yeast evenly on top of the milk and allow it to sit for approximately seven minutes. Next, incorporate half a cup of flour and two tablespoons of sugar into the mixture. Cover the bowl with plastic wrap and let it rise in a warm oven set to 100°F for 25 to 30 minutes, or at room temperature for 35 to 45 minutes until it becomes puffed.

Once the initial rise is complete, add the eggs, three tablespoons of melted butter, the remaining two tablespoons of sugar, salt, and vanilla extract to the mixture. Switch to the dough hook attachment and, on speed two, gradually mix in the remaining three cups of flour, adding half a cup at a time to allow for complete incorporation. The dough should feel slightly sticky but not adhere to the sides of the mixing bowl. If necessary, incorporate an additional one to two tablespoons of flour. Knead the dough on speed two for eight to ten minutes.

Cover the dough with plastic wrap and let it rise in a warm oven for one hour or at room temperature for two hours until it has doubled in size. While the dough is rising, prepare the chocolate ganache.

After the dough has risen, divide it into two equal portions. On a generously floured clean surface, place one piece of dough and roll it into a rectangle approximately 12 inches wide by 16 inches long. Spread one-third of the chocolate ganache evenly over the dough, then tightly roll it up starting from one of the shorter sides.

Wrap the rolled dough in plastic wrap and refrigerate for 15 minutes. Using a sharp knife, cut each chilled roll in half lengthwise, keeping the top half connected. Gently twist the two halves together a few times.

Prepare two loaf pans by buttering and lining them with parchment paper. Carefully transfer the twisted dough into the prepared pans. Cover the pans with plastic wrap and allow the dough to rise in a warm area for 30 to 45 minutes or at room temperature for one to one and a half hours, until it becomes puffy. Meanwhile, preheat the oven to 350°F.

Once the dough has risen, brush the tops of the loaves with the remaining tablespoon of melted butter. Bake in the center of the preheated oven for approximately 30 minutes, or until the loaves are golden brown. For added shine, brush the hot loaves with coconut oil or spread the remaining chocolate ganache over the top.