Combine broth, sherry, soy, garlic, ginger, and mushroom stems in a medium pot. Bring to a boil over high heat then reduce heat to low and simmer until broth is infused with flavors, about 30 minutes.
Strain broth, return to pot over high heat, add broth and bring to a boil. Reduce heat to low, add pork and mushrooms and cook until pork is just cooked through, about 3 minutes.
Garnish with green onions and sesame oil, if using, and serve.