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1
Combine the flour, yeast, sugar, and salt in the bowl of a stand mixer fitted with the dough hook. Add the eggs and milk and mix on low speed until a shaggy dough forms. Increase to medium speed and add the softened butter a few cubes at a time, waiting for each addition to be incorporated before adding the next. Continue mixing until the dough is smooth, supple, and pulls cleanly from the sides of the bowl. Cover the bowl and refrigerate overnight, or for a minimum of 8 hours. The dough is ready when it has doubled in volume and smells gently fermented.
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2
Melt the chopped dark chocolate with the butter in a heatproof bowl set over barely simmering water, stirring until smooth. Remove from the heat and sift in the cocoa powder and icing sugar, stirring to a uniform paste. Set aside at room temperature until the filling is spreadable but holds its shape when pressed — it must not be liquid. This is where the dish lives or dies: a filling that is too cold will tear the dough on spreading; a filling that is liquid will run from the loaf during shaping. The correct consistency is that of a thick, matte paste that falls from a spoon in a slow ribbon.
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3
Remove the dough from the refrigerator. On a lightly floured surface, roll it out to a rectangle approximately 40 cm × 30 cm. For a more complex interior, perform 4 to 6 letter folds before the final roll: fold the dough in thirds, rotate 90°, and roll out again, repeating. After the final roll, spread the chocolate filling across the entire surface in an even layer 3 to 4 mm thick, reaching to the edges. The dough is ready to be rolled when the filling covers every part of the sheet with no bare patches visible.
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4
Roll the dough tightly from the long edge into a compact log, pressing gently as you roll to avoid air pockets. Using a sharp knife, slice the log in half lengthwise from tip to tail in one clean cut, exposing the layered spiral cross-section. Twist the two halves around each other, cut sides facing outward, and transfer the twist into a lightly buttered Pullman tin or 900 g loaf tin. The shaped loaf is ready when it sits snugly in the tin with the spiral clearly visible along its length.
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5
Cover the tin loosely and leave the loaf to prove at room temperature until the dough is visibly puffed and springs back slowly — not immediately — when pressed lightly with a fingertip. Meanwhile, preheat the oven to 180 °C (conventional). Bake for 35 to 45 minutes, tenting with foil after the first 20 minutes if the top is browning too fast. The babka is done when an instant-read thermometer inserted into the thickest part of the filling reads 90 °C; the dense, wet filling requires this full temperature to be cooked through, and the loaf will feel firm and sound hollow when tapped on the base.
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6
While the babka bakes, bring the sugar and water together in a small saucepan over medium heat, stirring only until the sugar dissolves, then simmer for 2 minutes to form a light syrup. Remove the babka from the oven and brush the syrup generously over the entire surface immediately — the loaf must be hot for the syrup to absorb completely. Continue brushing until all the syrup is used. The surface should appear glossy and slightly tacky; a dull or dry surface indicates insufficient syrup.