Set one room temperature egg aside, then crack the second into a small bowl and set aside.
Combine flour, sugar, yeast and salt in the bowl of a stand mixer.
Add the warm milk and stir by hand or with the paddle attachment.
Crack in the 1 reserved egg, then scoop about half the yolk from the egg you cracked into bowl and add to the dough. Reserve the remaining cracked egg for your egg wash.
Add vanilla and softened butter to mixing bowl and stir to make a soft dough.
Attach the dough hook and allow the machine to knead the dough for 5 to 8 minutes or until smooth and shiny. The dough should be soft but not sticky and will clear the sides of the bowl, but stick to the bottom as it is being kneaded.
Scrape the kneaded dough into a buttered bowl, then cover and let rise for about an hour.
While dough is rising, grease and flour an 8 ½ by 4 ½ inch loaf pan. If desired, line lengthwise with a strip of parchment before flouring.
Punch down risen dough and place on a lightly floured pastry mat. Pat or roll into a 15x11 inch rectangle. The dough is so buttery you should be able to just pat it or roll it without using much flour at all. It should not be at all sticky. If it is, you'll need to use more flour on the mat.
Spread the hazelnut spread over the rectangle leaving about a ¾ inch border on the long sides. Sprinkle chocolate all over the hazelnut spread. Starting from a long side, use a cake lifter or thin spatula to lift edge of dough and roll carefully into a cylinder.
Cut the cylinder in half cross-wise so that you have two short tubes.
Cut one of the tubes lengthwise in half and lie the halves on their backs with chocolate exposed, pinch the top ends together and twist. Repeat with second tube of dough so that you have two twists.
Lay one twist in the pan with the exposed chocolate side up, then lay the second twist over it in the pan exposed side down.
Cover with a greased piece of plastic wrap and let rise for about 45 minutes.
Meanwhile, preheat the oven to 350.
Take your reserved egg and brush all over the babka, then sprinkle with coarse sugar if desired.
Bake for 45-55 minutes (shield with a piece of foil after about 40 minutes) until a skewer inserted in the middle comes out clean. Let the babka cool completely, then carefully remove from pan and slice.