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1
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.
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2
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.
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3
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.
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4
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.
