To prepare meatballs: Combine beef, eggs, panko, oregano, cumin, pepper and 1/4 teaspoon salt in a large bowl. Let stand for 10 minutes at room temperature.
Meanwhile, bring 2 quarts of water to a simmer in a large saucepan over high heat. Form the beef mixture into 32 balls, about 1 tablespoon each. Add half the meatballs to the simmering water and cook for 3 minutes. Use a slotted spoon to transfer them to a plate. Repeat with the remaining meatballs.
See AlsoLasagna SoupTo prepare soup: Heat oil in a large pot over medium-high heat. Add onion and garlic; cook, stirring, until fragrant, 1 to 2 minutes. Add chayote (or summer squash), potato and carrot; cook, stirring occasionally, until the vegetables begin to soften, about 3 minutes. Add tomato paste and cook, stirring constantly, until the vegetables are coated, about 1 minute. Add tomatoes and their juice; cook, stirring, for 1 minute. Add broth, zucchini, salt and the meatballs. Bring to a boil, then reduce heat to maintain a simmer. Cook, stirring occasionally, until the meatballs are cooked through, about 10 minutes. Remove from heat and stir in parsley.
Tips
To make ahead: Prepare through Step 2 and refrigerate meatballs for up to 1 day.
Tip: Chayote squash are small, pear-shaped squash with a mild, sweet taste. To peel and seed them, cut a thin slice off each end and use a vegetable peeler to remove the thin skin. With a sharp knife, quarter the squash, then slice off the white center and the seed. If you can't find chayote, use summer squash instead.
