Cheesy Ham Chowder

Name: Cheesy Ham Chowder
Ingredients: 4 tablespoons butter; 1 large onion, chopped; 3 carrots, diced; 2 stalks celery, diced; 2 cloves of garlic, chopped; 1/4 cup all-purpose flour; 4 cups low-sodium chicken stock; 2 cups milk; 2 large russet potatoes, peeled and cubed; 2 bay leaves; 1 1/2 pounds fresh broccoli, chopped; 1 pound ham steak (or leftover ham), chopped; Freshly grated nutmeg; Kosher salt; Fresh cracked black pepper; 1 cup half and half; 2 cups sharp yellow cheddar cheese, shredded; 1/2 cup Parmesan cheese, grated