Cheesy Vegetable Chowder

Name: Cheesy Vegetable Chowder
Ingredients: 5 1/2 tablespoons Butter (divided); 1 1/2 cups Carrots (shredded); 1 cup Celery (chopped); 1 cup Onion (chopped); 2 tablespoons Minced garlic; 3 cups Chicken broth; 4 cups Russet potatoes (peeled and diced); 3 cups Broccoli florets (chopped); 1/4 teaspoon Salt; 1/8 teaspoon Pepper; 1/4 teaspoon Dried parsley; 6 tablespoons All-purpose flour; 1 cup Half and half; 2 1/2 cups Milk; 2 cups Cheddar cheese (shredded); 1/2 cup Parmesan cheese (shredded)