For the Dough:
In the bowl of a stand mixer, combine warm milk, 1 tablespoon of sugar, and yeast. Let stand for 5 minutes until foamy.
Add the remaining sugar, flour, salt, eggs, egg yolk, and vanilla. Mix with the dough hook on low speed until combined, about 2 minutes.
Increase speed to medium and knead for 5 minutes until smooth.
With the mixer running, add softened butter one piece at a time, allowing each piece to incorporate before adding the next.
Continue kneading for 8-10 minutes until the dough is smooth, elastic, and pulls away from the sides of the bowl.
Transfer dough to a lightly greased bowl, cover with plastic wrap, and let rise in a warm place until doubled in size, about 1.5-2 hours.
For the Filling:
While the dough rises, pulse toasted hazelnuts with 1 tablespoon of sugar in a food processor until finely ground but not pasty.
In a heatproof bowl set over simmering water, melt chocolate and butter together, stirring occasionally until smooth.
Remove from heat and stir in sugar, cocoa powder, cinnamon, salt, and ground hazelnuts. Let cool until spreadable but not hot.
Shaping the Babka:
Punch down the risen dough and transfer to a floured surface. Roll out to a 16×12-inch rectangle, with the long side facing you.
Spread the chocolate-hazelnut filling evenly over the dough, leaving a ½-inch border along the edges.
Starting from the long edge closest to you, tightly roll the dough into a log.
Using a sharp knife, cut the log in half lengthwise, exposing the layers.
With the cut sides facing up, twist the two halves around each other, maintaining the exposed layers facing upward.
Carefully transfer the twisted dough to a 9×4-inch loaf pan lined with parchment paper.
Cover loosely with plastic wrap and let rise until puffy, about 45-60 minutes.
Baking:
Preheat oven to 375°F (190°C).
Brush the top of the babka with egg wash.
Bake for 30-35 minutes until golden brown and an instant-read thermometer inserted into the center reads 190°F.
For the Syrup:
While the babka is baking, combine water and sugar in a small saucepan. Bring to a simmer, stirring until sugar dissolves. Remove from heat.
As soon as the babka comes out of the oven, brush the hot syrup all over the top.
Allow the babka to cool in the pan for 15 minutes, then use the parchment paper to lift it out onto a wire rack to cool completely before slicing.