Place mushrooms in a small bowl and cover with water. Let soak until softened, about 15 minutes. Drain and cut into bite-size pieces.
Bring 1 1/2 cup water to a boil in a pot. Add pork and bamboo shots. Cook, skimming off any fat that rises to the top, until pork is tender, 5 to 10 minutes.
Mix sugar, sake, soy sauce, black rice vinegar, and chile paste together in a small bowl. Stir into the pot. Reduce heat to low and simmer soup, covered, about 10 minutes.
Stir mushroom pieces into the soup. Crack in eggs and cook, covered, until whites are firm and the yolks have thickened but are not hard, 2 1/2 to 3 minutes. Drizzle sesame oil over soup before serving.