Heat 1 tablespoon olive oil in a large pot over medium-high heat. Add the sausage and cook, stirring occasionally, until browned, about 4 minutes. Transfer to a bowl using a slotted spoon. Add the squash, carrots, celery, onion, chicken broth, 2 1/2 cups water and 3/4 teaspoon salt to the pot. Cover and bring to a simmer, then reduce the heat to low and continue simmering until the vegetables are tender, about 15 more minutes.
Puree the soup in a blender in batches, then return to the pot along with the sausage. Season with salt and pepper and keep warm.
Heat the remaining 1/3 cup olive oil in a small saucepan over medium-high heat. When the oil is very hot, add the sage leaves in batches and fry, turning, until dark green and crisp, about 1 minute. Remove with a slotted spoon and drain on paper towels. Ladle the soup into bowls and top with the fried sage.
Photograph by Christopher Testani
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.