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. With the mixer still on low, add the softened butter one tablespoon at a time. Increase to medium speed and knead for 8 to 10 minutes until the dough is smooth, elastic, and pulls away cleanly from the bowl. Transfer to a lightly greased bowl, cover with plastic wrap, and let rise in a warm spot until doubled in size, about 1.5 hours.
02 - Place the chopped dark chocolate and butter in a heatproof bowl set over a pot of barely simmering water. Stir gently until completely melted and smooth. Remove from heat and whisk in the powdered sugar, cocoa powder, and orange zest until well combined. Let the mixture cool for about 10 minutes until it thickens to a spreadable consistency.
03 - Punch down the risen dough to release air bubbles. On a lightly floured surface, roll the dough into a 16 by 12 inch rectangle. Spread the chocolate filling evenly across the surface, leaving a ½ inch bare border along all edges. Starting from one long side, roll the dough tightly into a log, pinching the seam to seal.
04 - Using a sharp knife, slice the log in half lengthwise down the center, keeping one end attached. Rotate the two halves so the cut sides face upward, then twist them around each other, tucking the ends underneath. Carefully transfer the twisted loaf into a greased 9 by 5 inch loaf pan. Cover loosely with a kitchen towel and let rise until puffy and nearly doubled, 45 to 60 minutes.
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 last 10 minutes.
06 - While the babka bakes, whisk together the powdered sugar, fresh orange juice, and orange zest until completely smooth. Allow the baked babka to cool in the pan for 10 minutes, then turn out onto a wire rack. Drizzle the orange glaze generously over the warm babka, letting it drip down the sides. Cool completely before slicing.