In a 6-quart heavy-bottomed soup pot, heat the oil over medium heat and stir in the onions and squash. Sauté until the onions begin to soften, about 3 minutes. Add 1 cup of stock, cover, reduce the heat to very low and sweat the vegetables for 20 minutes.
Add the remaining stock and thyme and bring to a boil. Cover, reduce the heat, and simmer until the squash is tender, about 30 minutes.
Remove from heat and allow to cool slightly. Using an immersion blender, puree the soup (if you do not have an immersion blender, you can use a food processor or blender, but you must wait for the soup to completely cool before you puree).
In a medium sauté pan, heat the remaining oil and sauté the sausage until brown and crisp, about 8 minutes. Keep warm.
Reheat the soup and season to taste with salt and pepper. Ladle the soup into shallow bowls. Spoon 3 tablespoons of sausage in the center of each bowl and top with a dollop of crème fraiche.