Simmer chayote slices in 2 cups of salted water in a medium saucepan until tender, about 20 minutes.
Sauté onion and garlic in a large, non-stick saucepan until the onion is soft. Add the shredded potato and stir for a minute. Pour in vegetable stock, bring to a boil, and cook over medium heat for about 15 minutes, until thickened.
Add the chayote and its cooking water to the stock, and using a hand (immersion) blender, puree the soup right in the pan. Stir and season to taste. Let simmer, uncovered, for about 5 more minutes. When ready to serve, ladle into bowls, sprinkle a few green onion slices on each and squeeze in some fresh lemon.