In a large mixing bowl, combine the warm milk, melted butter, and sugar. Sprinkle the yeast over the top and let it sit for about 5 minutes until frothy.
Add the egg and vanilla extract to the yeast mixture and stir until combined.
In a separate bowl, whisk together the flour, salt, and cocoa powder. Gradually add the dry ingredients to the wet ingredients, mixing until a dough forms.
Knead the dough on a floured surface for about 5-7 minutes until smooth and elastic. Place the dough in a greased bowl, cover with a clean towel, and let it rise in a warm place for about 1 hour or until doubled in size.
Once the dough has risen, punch it down and roll it out on a floured surface into a rectangle about 1/4 inch thick.
Spread the softened butter over the dough, then sprinkle the brown sugar, cinnamon, and chocolate chips evenly on top.
Starting from one long edge, roll the dough tightly into a log. Slice the log into 12 equal pieces and place them in a greased baking dish.
Cover the rolls with a towel and let them rise for another 30 minutes.
Preheat the oven to 350°F. Bake the rolls for 25-30 minutes or until golden brown.
While the rolls are baking, prepare the icing by whisking together the powdered sugar, milk, and vanilla extract until smooth.
Once the rolls are done, remove them from the oven and drizzle the icing over the warm rolls. Serve immediately.