Pancotto Tuscan Bread Soup

Name: Pancotto (Tuscan Bread Soup)
Ingredients: 1/2 pound (250 grams) stale white bread (Italian style loaf), crusts on; 3 tablespoons extra virgin olive oil, plus more for serving; 1/2 onion, finely chopped; 1 small carrot, finely chopped; 1/2 celery stick, finely chopped; 1 clove garlic, peeled but whole; 1 pinch salt; 1 teaspoon dried chile (or chopped fresh chile); 4 cups (1 liter) water or stock (vegetable or beef); 3/4 cup (80 grams) finely grated Pecorino Romano or Parmesan cheese, divided; Handful of fresh herbs, such as oregano, marjoram, or parsley