Preheat the oven to 450 degrees F. Heat 2 tablespoons olive oil in a large pot over medium-high heat. Stir in the onion and cook until softened, about 5 minutes. Stir in 3 minced garlic cloves, 1/2 teaspoon oregano and a pinch of salt; cook until softened, about 30 seconds. Add 1 cup water, the tomatoes, squash, chicken broth, parmesan rind, 3/4 teaspoon salt and a few grinds of pepper. Bring to a boil, then reduce the heat and simmer until the squash is very tender, 15 to 20 minutes.
Meanwhile, combine the remaining 3 tablespoons olive oil, 2 garlic cloves, 1 teaspoon oregano and a pinch of salt in a small bowl. Brush inside the bread bowls and place on a baking sheet. Cut the bread tops into cubes and add to the baking sheet. Bake until toasted, 8 to 10 minutes. Sprinkle the mozzarella around the edges of the bread bowls and on top of the bread cubes. Continue baking until the cheese starts melting, 2 to 3 minutes.
Stir the heavy cream into the soup, bring to a simmer and remove from the heat. Discard the parmesan rind. Working in batches, puree the soup in a blender. Serve the soup in the bread bowls. Top with the cheesy croutons and parmesan and drizzle with olive oil.