In the bowl of a stand mixer, combine the lukewarm milk, yeast, and 1 teaspoon of the granulated sugar. Stir gently and let sit for 5-10 minutes until foamy, indicating the yeast is active.

Add the remaining granulated sugar, flour, and salt to the yeast mixture. With the dough hook attachment, mix on low speed until just combined.

Add the eggs one at a time, mixing well after each addition. Add the vanilla extract. Continue to mix for 2-3 minutes until a shaggy dough forms.

With the mixer running on low, add the butter pieces one at a time, allowing each piece to incorporate before adding the next. This process will take about 5-7 minutes. Once all butter is added, increase speed to medium-low and knead for 8-10 minutes until the dough is smooth, elastic, and pulls away from the sides of the bowl. It will be soft and slightly sticky.

Lightly grease a large bowl. Transfer the dough to the bowl, turning once to coat. Cover tightly with plastic wrap and refrigerate for at least 4 hours, or preferably overnight (up to 24 hours). Chilling the dough makes it much easier to work with.

While the dough chills, prepare the filling. In a heatproof bowl set over a saucepan of simmering water (or in a microwave-safe bowl), melt the chopped dark chocolate and melted butter together until smooth. Stir occasionally.

Remove from heat and stir in the powdered sugar, cocoa powder, and salt until completely smooth and free of lumps. The filling should be a thick, spreadable consistency. Set aside to cool to room temperature; it will thicken further.

Lightly grease a 9x5 inch loaf pan and line with parchment paper, leaving an overhang on the longer sides for easy removal.

On a lightly floured surface, roll out the chilled dough into a large rectangle, approximately 12x18 inches (30x45 cm) and about 1/8 inch thick.

Spread the cooled chocolate filling evenly over the entire surface of the dough, leaving a small 1/2-inch border on one of the long edges.

Starting from the long edge opposite the clean border, tightly roll up the dough into a log. Pinch the seam to seal it.

Using a sharp knife, carefully slice the log lengthwise down the middle, creating two long halves, exposing the chocolate layers.

Place the two halves next to each other, cut-side up. Gently twist them together, keeping the cut sides facing up as much as possible, creating a braided effect. Pinch the ends together to seal.

Carefully transfer the twisted dough into the prepared loaf pan. Cover loosely with plastic wrap or a clean kitchen towel and let it proof in a warm place for 1 to 1.5 hours, or until visibly puffy and almost doubled in size.

Preheat your oven to 375°F (190°C).

Bake the babka for 30-40 minutes, or until deeply golden brown and a wooden skewer inserted into the center comes out clean (or with moist crumbs, not raw dough). If it browns too quickly, tent loosely with foil.

While the babka bakes, prepare the simple syrup: In a small saucepan, combine the granulated sugar and water. Bring to a boil over medium heat, stirring until the sugar dissolves. Remove from heat.

As soon as the babka comes out of the oven, immediately brush generously with the warm simple syrup. This adds moisture, shine, and extra sweetness.

Let the babka cool in the pan for 15-20 minutes before using the parchment paper overhang to lift it out onto a wire rack to cool completely. Slice and serve.