01 - In a stand mixer bowl, combine the flour, sugar, and yeast. Pour in the lukewarm milk, add the eggs, vanilla extract, and salt. Mix on low speed until a shaggy dough forms. Gradually add the softened butter one tablespoon at a time, then increase speed to medium and knead for 8–10 minutes until the dough is smooth, elastic, and pulls away cleanly from the bowl.
02 - Transfer the dough to a lightly greased large bowl, cover tightly with plastic wrap or a damp towel, and set in a warm, draft-free spot. Let rise until doubled in volume, approximately 1½ hours.
03 - Place the chopped dark chocolate and butter in a heatproof bowl set over a pot of barely simmering water (or microwave in 30-second bursts). Stir until completely melted and smooth. Whisk in the powdered sugar, cocoa powder, and orange zest until well blended. Allow the mixture to cool and thicken slightly for easier spreading.
04 - Punch down the risen dough and turn it out onto a lightly floured surface. Roll into a 16×12-inch rectangle. Spread the chocolate filling evenly across the dough, leaving a ½-inch border on all sides. Starting from the long edge, roll the dough tightly into a log. Using a sharp knife or bench scraper, slice the log lengthwise down the middle to create two halves. Twist the two halves together, keeping the cut sides facing up, and transfer the twisted loaf into a greased 9×5-inch loaf pan.
05 - Cover the pan loosely with a damp towel or oiled plastic wrap. Let the shaped babka rise for 45–60 minutes until noticeably puffy and nearly doubled.
06 - Preheat the oven to 350°F (175°C). Bake the babka on the center rack for 40–45 minutes, rotating halfway through, until the top is deep golden brown and a skewer inserted into the center comes out clean. Tent loosely with foil if browning too quickly.
07 - While the babka bakes, whisk together the powdered sugar, fresh orange juice, and orange zest until a smooth, pourable glaze forms. Let the babka cool in the pan for 10 minutes, then turn out onto a wire rack. Drizzle the orange glaze generously over the warm babka, allowing it to drip down the sides.