Activate the yeast: In a small bowl, combine the warm milk, yeast, and 1 tablespoon of the sugar. Let stand for 5-10 minutes until foamy.
Mix the dough: In a large bowl or stand mixer, combine the remaining sugar, eggs, vanilla, salt, and flour. Pour in the yeast mixture. Mix on low speed until a shaggy dough forms.
Knead: Add the softened butter pieces one at a time, mixing until fully incorporated. Increase speed to medium and knead for 8-10 minutes until the dough is smooth and elastic.
First rise: Place the dough in a lightly oiled bowl, cover, and let rise in a warm place for 1.5 to 2 hours, or until doubled in size.
Prepare the filling: While the dough rises, melt the 8 ounces of chopped chocolate. In a separate bowl, beat the 1 cup softened butter with the powdered sugar and cocoa powder until smooth. Mix in the melted chocolate until combined.
Shape the dough: Punch down the risen dough. Turn it out onto a lightly floured surface and roll it into a large rectangle, about 12×18 inches.
Spread the filling: Spread the chocolate filling evenly over the entire surface of the dough, leaving a 1-inch border on one long edge.
Roll and slice: Starting from the long edge opposite the border, tightly roll the dough into a log. Using a sharp knife or bench scraper, slice the log lengthwise down the center, exposing the layers.
Braid: Turn the two halves cut-side up. Cross the strands over each other, keeping the cut sides facing up as much as possible to expose the chocolate swirls. Pinch the ends together.
Bake: Carefully transfer the braid to a greased 9×5 inch loaf pan. Cover loosely and let rise for 45 minutes. Preheat your oven to 350°F (175°C). Bake for 35-45 minutes, or until golden brown.
Make the syrup: While the babka bakes, combine the water and 1/4 cup sugar in a small saucepan. Heat until the sugar dissolves.
Glaze: Immediately upon removing the babka from the oven, brush the hot syrup evenly over the top and sides. Let cool in the pan for 15 minutes before transferring to a wire rack to cool completely.
Notes
For a softer dough, use bread flour instead of all-purpose flour.
If the dough is too sticky during rolling, chill it for 15 minutes before proceeding.
You can make the dough a day ahead and let it cold-proof in the refrigerator overnight for better flavor development.