In the bowl of a stand mixer, stir together the warm milk and half of the sugar. Sprinkle the yeast over top of the milk and allow the yeast to activate, about 5 minutes. The mixture should froth and foam slightly. You can stir it gently to make sure all the yeast has been moistened, but if the yeast does not foam, dump it out and start over. Once the yeast has been activated, stir 1/2 cup of the flour in to the milk mixture. Once combined, add the remaining sugar, 1 egg, and salt, and stir to combine. On low speed, add the remainder of the flour and beat to combine, scraping the sides of the bowl as needed. Once evenly combined, increase the speed to medium (I use speed number 4 on my Kitchen Aid stand mixer) and add the softened butter 1 tablespoon at a time. Scrape the sides of the bowl and then continue to beat on medium speed for an additional 4 minutes. The dough will be quite moist and sticky.
Lightly grease a large bowl and place the dough inside, covering it tightly with a piece of plastic wrap. Allow the dough to rest in a warm spot in the kitchen until it has approximately doubled in size, about 1-1/2-2 hours.
Once the bread has nearly risen, melt the chocolate, butter, and cocoa in a saucepan over low heat. Once smooth, stir in the powdered sugar until well combined. Set aside until the dough is ready. If it firms up beyond a spreadable consistency, gently melt it again.
Once the dough has risen, roll it out into a 12”x18” rectangle on a lightly floured surface. Spread the chocolate filling evenly over the rectangle, leaving a 1” border around the perimeter of the dough. Sprinkle evenly with the peppermint bark bits.
Starting with one of the long ends, roll the dough somewhat tightly into one long tube and pinch the ends of the dough to seal in the filling. Use a sharp knife to slice down the middle of the length of the dough, leaving one inch at the top uncut. You should end with two strands of rolled dough connected by a 1” chunk of dough at the top. This will serve as the starting point for your braid, so you want the dough to remain connected here. Turn the exposed innards of the dough roll to face upwards and then twist the dough, right over left, right over left, until the entire length of dough has been twisted into one long piece. Join the ends of the dough to make a ring, pinching the ends together slightly. Carefully transfer the ring of dough into a lightly greased 10” cast-iron skillet. Cover the skillet with a piece of plastic wrap and allow the dough to rest while you preheat the oven to 350 degrees.
Once the oven is preheated, stir the remaining egg with 1 tablespoon of water and brush a thin layer of the mixture over the surface of the dough. Bake in the preheated oven for 25-30 minutes, or until the dough is puffed and golden brown throughout. Allow to cool slightly prior to cutting and serving. Feel free to dust with powdered sugar once cooled.