In a food processor with the fine shredder blade, process squash and potatoes. In a six-quart (five- and- a- half-liter) pot, sauté onion in oil. Add vegetables and stir until vegetables have softened. Add water to cover.

Bring soup to a boil; reduce heat and cook 10 minutes. Add cauliflower and more water to cover vegetables. Add remaining ingredients and return to a boil. Reduce heat; partially cover pot and cook additional 15–20 minutes, or until all vegetables are soft.