Combine the lukewarm milk, 1 tablespoon of sugar, and yeast in a large bowl. Whisk to dissolve the yeast. Leave for 5-10 minutes until foamy.
Add in the eggs, remaining sugar, vanilla, and salt. Add the flour. Mix until smooth.
Slowly incorporate the butter. Knead the dough with the hook of your mixer or by hand for 5-8 minutes until nice and smooth.
Place the dough in a greased bowl. Cover the dough with plastic wrap and let rise in a warm place for one hour, or until doubled in size.
For the filling, melt the chocolate in the microwave, stirring at 30-second intervals until smooth. Mix in the butter, cocoa powder, and powdered sugar until a homogeneous mixture forms.
Grease a 9x5 loaf pan. Set aside.
Place the dough on a lightly floured surface. Knead for a short while. Roll the dough out to a large rectangle. Spread the chocolate filling, leaving the dough bare around the edges.
Roll up the dough tightly to form a log. Using a sharp knife, cut the log longwise, leaving the last inch of the dough intact on one end. Twist/braid the two sections of dough together.
Carefully transfer to prepared loaf pan. Cover loosely with plastic, and let rise for 20 minutes.
Meanwhile, preheat oven to 325°F.
Brush the top of the loaf with the egg wash.
Bake for 30-35 minutes until golden brown. Cover with foil, if needed. Allow the loaf to cool in the pan for 15 minutes, then transfer to a cooling rack.
To make the glaze, put the chocolate and butter in a microwave-safe bowl and microwave for 30-second intervals, or until the chocolate and butter are melted and smooth. Add in the corn syrup. Beat the mixture, adding small amounts of hot water until thinned as desired.
Pour or spoon the glaze over the cooled chocolate babka.
This recipe yields 1 loaf.
Another popular version of babka includes cinnamon in the filling, so feel free to add a bit if you would like. Baking chocolate can be used in place of chocolate chips if preferred, and you can certainly use dark chocolate in the semi-sweets place.