MAKE THE STARTER: About 8 to 12 hours before mixing your dough, combine the starter, water, and flour in a bowl. Mix well, cover, and leave at room temperature to activate. It will become bubbly and double in volume.
WARM THE MILK: Heat the milk until it reaches a temperature of 115°F to 120°F.
WEIGH THE INGREDIENTS: Making sure to tare your mixing bowl on the scale after each addition, combine 110 grams of warm milk, 70 grams of water, the sugar, and the salt. Then, the all-purpose flour, the whole-wheat flour, and 200g active starter
KNEAD: Mix to combine, once it becomes difficult to mix begin to knead. Turn the dough out and knead for 7 to 10 minutes by hand or 3 to 5 minutes on low speed in a stand mixer, until the dough is smooth, is no longer sticky, and releases easily from the bowl or work surface.
ADD BUTTER: Add the softened butter and knead in until it is completely incorporated. The dough should be smooth, elastic, and not greasy.
BULK FERMENT: Cover the dough and allow to ferment for 3 to 7 hours, until doubled in volume.
PREPARE THE LOAF PAN: Line a loaf pan with parchment paper and set aside.
SHAPE: On a lightly floured work surface, roll the dough with a rolling pin into an 11-by-18-inch rectangle. Spread the butter over the dough with a spatula until completely covered. Evenly sprinkle the mozzarella and Parmesan cheeses over the butter, then sprinkle the chives, rosemary, and parsley evenly over the cheeses. Starting at the short end, roll the dough up into a log. The final rolled dough will have lots of spiraled layers and will be 11 inches wide. Use a bread knife to cut through the log from end to end, splitting it in half longways. The lines of rolled dough, cheese, and herbs will show inside the cut. Carefully intertwine the two pieces two or three times, making sure that the layers of dough, cheese, and herbs face up. Handle the dough gently to help prevent the cheese and herbs from falling out. Carefully transfer the twisted dough into the parchment paper-lined loaf pan with the exposed layers facing up. Add extra mozzarella and Parmesan cheese to the top, if desired.
PROOF: Cover and proof in a warm place for 1.5 to 3 hours, until the layers appear puffy.
PREHEAT: Preheat the oven to 350°F.
BAKE: Place the proofed babka in the preheated oven. Bake for 40 to 50 minutes, until the cheese is melted and the top has a golden, toasted look.
COOL: Let cool for at least 20 minutes.