1. Prepare the Sourdough Dough Base: In a large mixing bowl, combine the warm milk, active sourdough starter, 1/4 cup granulated sugar, egg, and vanilla extract. Whisk until well combined. Gradually add 2 cups of the all-purpose flour and the salt, mixing with a wooden spoon until a shaggy dough forms.

2. Knead in the Remaining Flour and Butter: Turn the dough out onto a lightly floured surface. Knead for about 5 minutes, then gradually incorporate the softened 1/2 cup unsalted butter, a tablespoon at a time, until fully absorbed. Continue kneading, adding the remaining 1 cup of flour as needed, until the dough is smooth, elastic, and no longer sticky (about 8-10 minutes of total kneading). It should pass the windowpane test (stretch a small piece until it’s translucent without tearing).

3. First Rise (Bulk Fermentation): Place the kneaded dough in a lightly oiled bowl, turning once to coat. Cover the bowl tightly with plastic wrap or a damp kitchen towel. Let it rise in a warm place for 2-3 hours, or until doubled in size. The dough should look puffy and slightly bubbly.

4. Prepare the Chocolate Filling: While the dough is rising, melt 2 tablespoons of unsalted butter in a small saucepan or microwave-safe bowl. In a separate bowl, combine the semi-sweet chocolate chips, cocoa powder, and 2 tablespoons of granulated sugar. Pour the melted butter over the chocolate mixture and stir until the chocolate chips are mostly melted and the mixture forms a thick, spreadable paste. If needed, gently warm it further over a double boiler or in short microwave bursts, stirring frequently, until smooth.

5. Shape the Babka Rolls: Once the dough has doubled, gently punch it down and turn it out onto a lightly floured surface. Roll the dough into a large rectangle, approximately 12×18 inches. Spread the chocolate filling evenly over the entire surface of the dough, leaving a small border along one long edge.

6. Roll and Slice: Starting from the long edge opposite the border, tightly roll the dough into a log. Use a sharp knife or unflavored dental floss to cut the log into 12-14 equal-sized rolls, about 1.5 inches thick.

7. Second Rise and Baking: Arrange the cut rolls in a greased 9×13-inch baking dish, leaving a little space between each. Cover loosely with plastic wrap or a damp towel and let them rise again in a warm place for 30-45 minutes, or until visibly puffy. Preheat your oven to 375 F (190 C) during the last 15 minutes of the second rise. Bake for 20-25 minutes, or until the tops are golden brown and the rolls are cooked through.

8. Cool and Serve: Remove the babka rolls from the oven and let them cool in the baking dish for about 10 minutes before transferring them to a wire rack. Dust with powdered sugar before serving warm or at room temperature.