Heat the oil in a large Dutch oven or stockpot over medium heat until shimmering. Add the onion, season with salt, and cook, stirring occasionally, until softened, about 5 minutes. Add the ginger, garlic, and chile and cook until fragrant, about 30 seconds.
Add 2 tablespoons of the curry paste, fish sauce, and brown sugar and cook until fragrant and the brown sugar is completely dissolved, about 30 seconds. Add the coconut milk, stir to combine, and bring to a simmer. Taste and add more curry paste as needed — you want the sauce to taste strong and concentrated, as the mussels will release liquid and dilute it as they cook. (See Recipe Note.)
Add the mussels, stir to combine, and cover the pot with a tight-fitting lid. Cook covered, stirring the mussels every minute or so, until the mussels are opened, 5 to 8 minutes total. Transfer the mussels to serving bowls, discarding any unopened mussels. Pour the broth over the mussels and sprinkle with cilantro. Serve with lime wedges and bread, noodles, or rice.