Heat the oil in a large saucepan over medium-high heat. Add the scallion whites, garlic, and ginger and cook, stirring, until fragrant, about 30 seconds. Add the pork and cook, breaking it up with a spoon, until browned and cooked through, 6 to 8 minutes. Add the mushrooms and 1/2 teaspoon salt and cook, stirring frequently, until tender, 2 to 3 minutes more.
Add the broth, fish sauce, and 2 cups water to the saucepan and bring to a boil. Add the wonton wrappers, reduce heat, and simmer until tender, 1 to 2 minutes. Stir in the lime juice. Serve with the scallion greens, chili peppers, cilantro, and lime wedges.