In a microwavable Pyrex measuring cup, heat the water to 100-110 degrees F. Use a thermometer to be sure, because anything too hot will kill the yeast in the next step.
Once the water is no more than 110 degrees, add a large pinch of sugar to the water, and the yeast. Stir to mix, and let it sit for 5-10 minutes or until foamy.
In the meantime, carefully measure out the activated charcoal and the flour into a large mixing bowl.
Once the yeast is foamy, add the egg, oil, remaining sugar, and salt. Put the wet ingredients into the dry.
Use a rubber spatula to mix until thoroughly combined.
Once a dough has formed, turn it out onto a large cutting board that has been dusted with flour. Coat your hands with flour and knead for 3 minutes. Add in more flour if the mixture is overly sticky.
Divide into four equal parts and form into a round ball.
Cover and let rest in a warm area for 10 minutes. Meanwhile, preheat the oven to 425.
After the buns have risen, lightly beat together the water and egg while in a small bowl.
Brush the water-egg mixture lightly on the tops of the buns, one at a time, adding black sesame seeds immediately after brushing.
Bake for 12 minutes.
Cool on wire racks and cut in half crosswise when they are cool enough to handle.