Put oil in a deep saucepan over medium-high heat. Add onion and cook, stirring occasionally, until soft, 2 or 3 minutes. Add rice and stir to coat with oil, then continue cooking, stirring occasionally, until fragrant, about 2 minutes.
Add zucchini along with a light sprinkle of salt and a few grinds of pepper. Stir constantly for a couple of minutes, until zucchini starts to wilt and release its liquid; add about half the parsley. When mixture starts to stick to bottom of pan, stir in 6 cups water. Bring soup to a boil, and reduce heat so mixture simmers steadily. Cover and cook for 20 to 30 minutes, until rice is tender and vegetables start to melt into soup.
Beat eggs in a 4-cup or larger heat-resistant bowl, then whisk in the lemon juice. Take a ladle of broth from pot (be careful not to include too many vegetables) and slowly add broth to eggs, a few drops at a time at first, whisking constantly so eggs do not curdle. Repeat once or twice more, until egg mixture is thick, smooth and very warm.
Adjust heat so that soup bubbles gently. Slowly add egg mixture, stirring constantly. Taste and adjust seasoning, adding more lemon juice, salt and/or pepper as needed. Serve immediately, garnished with remaining parsley and, if you like, cheese.