Bring a pot of water to a boil for the noodles. Meanwhile, in a large pot or Dutch oven, heat the vegetable oil over medium-high. Add the squash, season with salt and cook, stirring occasionally until slightly softened and lightly golden in spots, 5 to 7 minutes. Stir in the curry paste, ginger and turmeric. Cook, stirring often, until very fragrant, about 1 minute, lowering the heat if necessary to keep from scorching.
Add the coconut milk and bring to a simmer. Cover and cook, stirring occasionally, until squash is tender enough to mash, 10 to 12 minutes. Uncover and use a wooden spoon or potato masher to crush the squash into small pieces. Add 2 cups of the stock, fish sauce and brown sugar. Bring to a boil, then turn the heat off. Add more stock for a brothier soup. Season with salt and adjust other seasonings to taste.
While the squash simmers, cook the noodles per the package directions.
To serve, divide the noodles into bowls, ladle the soup over, very generously squeeze lime over, and top with cilantro, shallots and chile oil.