01 - In a stand mixer bowl, combine the all-purpose flour, granulated sugar, and instant dry yeast. Pour in the lukewarm whole milk, add the room-temperature eggs, vanilla extract, and salt. Mix on low speed until a shaggy dough forms. Gradually add the softened unsalted butter, one tablespoon at a time, then knead for 8 to 10 minutes until the dough is smooth, glossy, and elastic. Transfer to a lightly greased bowl, cover with plastic wrap, and let rise in a warm spot until doubled in size, approximately 1½ hours.
02 - Place the chopped dark chocolate and unsalted butter in a heatproof bowl set over a pot of barely simmering water. Stir gently until completely melted and smooth. Whisk in the powdered sugar, unsweetened cocoa powder, and orange zest until well blended. Allow the filling to cool slightly so it thickens to a spreadable consistency.
03 - Gently punch down the risen dough to release air bubbles. On a lightly floured surface, roll the dough out into a 16×12-inch rectangle. Spread the chocolate filling evenly across the surface, leaving a ½-inch border along the edges. Starting from one long side, roll the dough tightly into a log. Using a sharp knife, slice the log in half lengthwise. Twist the two halves together with the cut sides facing up, then carefully transfer the twisted loaf into a greased 9×5-inch loaf pan.
04 - Cover the loaf pan loosely with a damp kitchen towel or plastic wrap. Let the shaped babka rise in a warm, draft-free area for 45 to 60 minutes until visibly puffy and nearly doubled.
05 - Preheat the oven to 350°F. Bake the babka on the center rack for 40 to 45 minutes until the top is deep golden brown and a skewer inserted into the center comes out clean. If the top browns too quickly, tent loosely with aluminum foil during the final 10 minutes.
06 - While the babka bakes, whisk together the powdered sugar, fresh orange juice, and orange zest in a small bowl until smooth and pourable. Adjust the consistency with additional juice or sugar as needed.
07 - Let the baked babka cool in the pan for 10 minutes, then turn it out onto a wire cooling rack. While still warm, drizzle the orange glaze generously over the top, allowing it to drip down the sides. Let cool completely before slicing into 10 to 12 portions.