To Make the Broth:

Heat the oven to 450°F.

Combine all of the ingredients for the broth, except the water and peppercorns, in a large roasting pan and toss well to combine. Place the pan in the oven and roast the vegetables for 30 to 40 minutes, turning every 15 minutes or so, until they are soft and browned in places.

Transfer the vegetables and their juices to a large pot and pour in the water. Add the peppercorns and bring the broth to a boil over medium-high heat. Reduce the heat to medium-low, cover partially, and cook for 45 minutes, until the broth is richly flavored.

Remove the broth from the heat and let sit for 5 minutes. Using a fine-mesh sieve lined with damp cheesecloth, strain the broth into a clean pot. Discard the solids. Return the broth to the burner, cover, and keep warm over low heat.

To make the soup:

In a large Dutch oven or other heavy-bottomed pot, warm the olive oil over medium heat. Add the carrots, fennel, and scallions and sauté, stirring frequently, for 10 minutes, until the carrots are bright and shiny and the scallions are softened. Add the zucchini and yellow squash and season with 1 teaspoon of salt. Cook for 3 minutes, just until the squashes start to soften.

Add the chickpeas and corn kernels and cook for a minute or two, then slowly pour in the warm broth. Add the meatballs. Bring the broth to a boil and stir in the shells. Cook for about 10 minutes or until the shells are al dente (cooking time will depend on the brand and type of pasta you use). Turn off the heat and add the spinach, stirring until it is wilted. Cover the pot and let the soup sit for just a minute or two. Stir in the basil and the cheeses.

Ladle the soup into individual shallow rimmed bowls and drizzle each serving with a little olive oil. Serve immediately.