Dissolve the yeast in the warm milk with a teaspoon of sugar in a large bowl. Let it sit for about 5 minutes. If the yeast is good, it will start to froth up.
To the bowl of your mixer add the ½ cup of sugar, butter, salt, eggs, and flour. Next, pour in the yeast mixture and mix well until well incorporated and the dough comes clean from the side of the bowl.
Place the dough into an oiled bowl, cover and let rise in a warm place about 1 hour or until the dough has doubled in size.
In a medium bowl, mix the softened butter, heavy cream, brown sugar, cocoa powder, and instant coffee powder until well combined. Fold in the chopped dark chocolate.
After the dough has risen, roll it out into a large rectangle (about 12x18 inches). Spread the filling evenly over the dough, leaving a small border around the edges.
Roll the dough tightly from the long edge into a log. Slice into 8 even rolls and place them in a greased 9x13-inch baking dish. Cover the dish with a clean kitchen towel and let the rolls rise for 30-45 minutes, or until they've doubled in size.
After the rolls have risen, pour ½ cup of heavy cream over them right before baking.
Preheat your oven to 350°F (175°C). Bake the rolls for 25-30 minutes, or until golden brown and cooked through.
While the rolls bake, whisk together powdered sugar, instant coffee powder, and heavy cream until smooth. Adjust the consistency with more cream if needed.
Once the rolls are baked and slightly cooled, drizzle the coffee icing generously over the top.