In a small bowl, combine warm milk, yeast, and 1 tablespoon of sugar. Let it sit for about 5-10 minutes until frothy.
In a large mixing bowl, combine the remaining sugar, softened butter, egg, and vanilla extract. Mix until smooth.
Add the frothy yeast mixture to the butter mixture and stir to combine.
Gradually add the flour and salt, mixing until a dough forms. Knead the dough on a floured surface for about 5-7 minutes until smooth and elastic.
Place the dough in a greased bowl, cover with a damp cloth, and let it rise in a warm place for about 1 hour or until doubled in size.
While the dough is rising, prepare the chocolate filling by mixing the chocolate chips and cocoa powder in a bowl.
Once the dough has risen, punch it down and roll it out on a floured surface into a rectangle (about 12x18 inches).
Sprinkle the chocolate filling evenly over the dough, leaving a small border around the edges.
Roll the dough tightly from the long side into a log. Pinch the seams to seal.
Cut the log in half lengthwise to expose the filling. Twist the two halves together and place in a greased loaf pan.
Cover the loaf pan with a cloth and let it rise for another 30-45 minutes.
Preheat the oven to 350°F. Bake the babka for 30-35 minutes until golden brown.
Allow it to cool slightly before dusting with powdered sugar. Slice and serve.