Wash the zucchini, trim the ends, and cut into medium chunks. No need to peel if the skin is tender.
Peel and dice the potato into pieces about the same size as the zucchini.
Peel and finely chop the onion.
In a medium pot, heat the extra-virgin olive oil over medium heat.
Add the chopped onion and sauté for 5–7 minutes, until translucent.
Stir in the zucchini and potato and cook for about 5 more minutes.
Pour in the water or vegetable broth, then season with salt and freshly ground black pepper.
Bring to a boil, reduce heat to low, cover, and simmer for 15–20 minutes, until the vegetables are tender.
Remove from heat and puree with an immersion blender until smooth.
If the soup is too thick, stir in additional water or broth until it reaches your desired consistency.
Optional: stir in the milk and blend again until fully combined.
Serve hot with a drizzle of olive oil, croutons, and fresh parsley leaves.