In a small bowl, combine warm milk and yeast. Let sit for 5 minutes until foamy.
In a large bowl, mix flour, sugar, and salt. Add yeast mixture, eggs, and 0.5 cup softened butter. Knead dough until smooth and elastic, about 10 minutes.
Cover dough and let rise in a warm place for 1 to 1.5 hours or until doubled.
Mix chopped dark chocolate, cocoa powder, powdered sugar, and remaining softened butter to create filling.
Roll out dough into a rectangle. Spread chocolate filling evenly over dough.
Roll dough tightly and cut lengthwise down the center, twisting the two halves to create the babka shape.
Place babka in a greased loaf pan and let rise for 30 minutes.
Bake at 350°F (175°C) for 40 minutes or until golden brown and cooked through. Cool before slicing.
Notes
For best results, use high quality dark chocolate and serve babka slightly warm with coffee or tea.