In a large heavy-bottomed pot, heat the olive oil over medium heat. Add the thinly sliced onions and a pinch of salt. Cook, stirring occasionally, until the onions become caramelized and golden brown, about 25 minutes. This slow cooking releases the natural sweetness of the onions, forming the flavorful base of the soup.

Once the onions are caramelized, add the minced garlic and cook for an additional minute until fragrant. Then, incorporate the vegetable scraps you've collected. These can include carrot peels, celery leaves, herb stems, and any other aromatic vegetable remnants. Sauté for about 5 minutes, allowing the scraps to soften and release their flavors.

Pour in the vegetable broth, ensuring all the ingredients are submerged. Add the bay leaf and dried thyme. Bring the mixture to a boil, then reduce the heat to low. Let the soup simmer gently for 20 minutes, allowing the flavors to meld together beautifully.

After simmering, taste the soup and adjust the seasoning with salt and freshly ground black pepper as needed. Remove and discard the bay leaf.

While the soup is simmering, preheat your oven to 350°F (175°C). Place the slices of stale Italian bread on a baking sheet and toast them in the oven until they're crisp and golden. To serve, ladle the hot soup into bowls, place a toasted bread slice on top of each, and generously sprinkle with shredded Parmesan cheese. For an authentic touch, you can place the bowls under a broiler for a minute to melt and slightly brown the cheese.