Heat oil in a large pot over medium heat. Add squash, carrots, and shallots, and season with salt. Cook, occasionally stirring, until the vegetables begin to soften, about 5 minutes.
Mix in broth, apple juice, and brown sugar. Turn the heat to high and bring to a boil, then reduce to a simmer and cover. Cook until vegetables are soft, about 20 minutes.
If you are using an immersion blender: blend the soup until smooth. Season with salt.If you are using a blender: Let the soup cool slightly. Add soup to your blender. Only fill it ¾ of the way full. Work in batches if necessary. Blend on high till the soup is smooth.