In a large bowl, mix together the beef, veal, breadcrumbs and 2 tbsp. water. Using a fork, stir in the egg, parsley and 1/2 cup parmesan; season generously with salt and pepper. Form the mixture into sixty 1/2-inch balls and set them on a platter.
In a large nonstick skillet, heat the vegetable oil over medium-high heat. Add the meatballs in 3 or 4 batches and cook, occasionally shaking the pan, until browned, 3 to 4 minutes. Using a slotted spoon, transfer to paper towels to drain.
In a large dutch oven or soup pot, bring the chicken broth, 2 cups water and the carrots to a boil over medium-high heat. Lower the heat and simmer for 5 minutes. Add the meatballs and return the soup to a simmer. Stir in the spaghetti and cook, stirring often, until just tender, 10 to 12 minutes. Stir in the spinach, cover and cook until wilted, about 5 minutes. Ladle the soup into bowls and top with the remaining 1/2 cup parmesan; serve immediately.