In a large bowl, mix the flour, sugar, yeast, and salt. Add the warm milk and yogurt and mix into a rough dough for 2 minutes.
Add the softened butter 1 tablespoon at a time while kneading, then keep kneading for 8-10 minutes total until the dough is smooth, elastic, and slightly tacky. Place in a greased bowl, cover, and let rise in a warm spot for 1 to 1 1/2 hours, until doubled.
Make the filling: melt the chocolate and butter together, whisk in the cocoa powder and powdered sugar, and cool for 10-15 minutes until thick and spreadable.
Roll the dough into a 10x16 inch rectangle on a floured surface and spread the filling evenly, leaving a 1/2 inch border.
Roll into a tight log from the long side, pinch the seam closed, and freeze for 15 minutes for cleaner cuts.
Cut the chilled log in half lengthwise so the chocolate layers are exposed. With the cut sides facing up, twist the two halves around each other and tuck into a parchment lined 9x5 inch loaf pan.
Cover and rise for 45-60 minutes until puffy. Preheat your oven to 350°F during the last 20 minutes.
Bake for 40-45 minutes, tenting with foil around the 25 minute mark if the top browns too quickly.
Simmer the sugar and water for 2 minutes, brush the syrup all over the hot babka, cool in the pan for 20 minutes, then move to a rack. Slice when just barely warm.
