Make the roux: in a small pot over medium heat, whisk together the milk and flour until it thickens into a paste, about 3 minutes. Set aside.
Dissolve the yeast: combine the yeast, a sprinkle of sugar and warm water in the bottom of a stand mixer bowl and set aside for 5 minutes. It will become foamy.
Make the dough: Over the yeast add the remaining dough ingredients and the roux. Knead on low then medium for 10-15 minutes, until the dough comes together around the hook. It will still be sticky and mostly shapeless but you’ll note that as it kneads it holds together better. Set in an oiled boil and cover with plastic wrap to rise for about 2 hours, until it has doubled in size.
Make the filling: Have the chopped chocolate nearby in a heatproof bowl and brown the butter by cooking it over medium heat in a frying pan until it has brown bits at the bottom and stops sputtering. Immediately pour over teh chocolate. Stir so the chocolate fully melts. Add the powdered sugar and cocoa, a pinch of salt and a dash of vanilla and stir until smooth.
Shape the dough: Divide the dough into two equal halves. Have two standard size loaf pans greased and lined with parchment paper ready. Roll one half out so that it’s a couple inches longer than your loaf pan and as wide as you can get it (at least 12-13”). Spread the filling over the dough, leaving a 1” border and sprinkle ⅓ cup of halva over it. Roll it up into a log, slice off the edges and then slice lengthwise so you have two long logs. Twist them around each other and set the babka in the loaf pan. Repeat with other dough half.
The second rise: Set the pans in a warm area for a second rise, about an hour but check for readiness by pressing the dough with a finger: if it springs back without leaving an indentation it needs more time. Preheat the oven to 350 F.
Bake: Brush the babkas with eggwash and sprinkle with sugar if you like. Bake for 35-40 minutes until the inside registers 190F. If the babkas are browning too quickly, tent with foil.