Heat oil in a large saucepan or medium Dutch oven over medium-high. Add mushrooms and cook, stirring occasionally, until browned, 5 to 8 minutes. Reduce heat to medium and stir in shallots, garlic, ginger, chile, and salt; cook, stirring frequently, until shallots are softened, 1 to 2 minutes. Stir in curry paste, and cook, stirring constantly, until slightly darkened and very fragrant, about 1 minute.
Stir in broth, coconut milk, fish sauce, lime juice, and brown sugar; bring to a simmer over medium-high.
Add dumplings, return to a simmer over medium-high, then stir in bok choy; reduce heat to medium. Simmer, uncovered and stirring occasionally, until dumplings are cooked through and bok choy is tender-crisp, 3 to 5 minutes.
Divide soup evenly between 4 bowls. Garnish with cilantro, Thai basil, sliced scallions, sliced red chile, crushed peanuts, and lime wedges. Serve immediately.
