Combine warm milk, yeast, and 1 teaspoon of the granulated sugar in a large bowl. Let stand for 5 minutes until foamy.
Stir in the melted butter, remaining granulated sugar, and egg into the yeast mixture.
Add the flour and salt. Mix until a shaggy dough forms, then knead on a lightly floured surface for 5-7 minutes until smooth and elastic.
Place the dough in a lightly buttered bowl, cover, and let rise in a warm place for 1 to 1.5 hours, or until doubled in size.
While the dough rises, prepare the filling: In a small bowl, mix the softened butter, brown sugar, cocoa powder, and cinnamon until smooth.
Punch down the risen dough. Roll it out on a lightly floured surface into a 12×18 inch rectangle.
Spread the chocolate filling evenly over the dough, leaving a 1-inch border on one long edge. Sprinkle the chopped chocolate over the filling.
Tightly roll the dough up starting from the long edge opposite the border. Pinch the seam closed.
Cut the log into 12 equal rolls. Place them cut-side up in a greased 9×13 inch baking pan. Cover and let rise for 30 minutes.
Preheat your oven to 375 degrees Fahrenheit. Bake for 20-25 minutes until golden brown.
While the rolls bake, make the frosting: Beat the cream cheese, powdered sugar, and vanilla until smooth. Add milk one tablespoon at a time until you reach a spreadable consistency.
Remove rolls from the oven and immediately spread the frosting over the warm rolls. Serve warm.
Notes
For extra gooey rolls, slightly underbake them by 2 minutes.
If you prefer a thinner glaze instead of thick frosting, use only 1/4 cup cream cheese and increase the milk to 3-4 tablespoons.
You can make the dough ahead of time and let it cold-proof in the refrigerator overnight for deeper flavor.