Cheesy Vegetable Chowder

Butter (5 1/2 tbsp (divided))

Carrots (shredded, 1 1/2 cups)

Celery (chopped, 1 cup)

Onion (chopped, 1 cup)

Garlic (minced, 2 tbsp)

Chicken broth (3 cups)

Russet potatoes (peeled and cubed, 4 cups)

Broccoli florets (chopped, 3 cups)

Salt (1/8 tsp)

Pepper (1/8 tsp)

Parsley (dried, 1/4 tsp)

All-purpose flour (6 tbsp)

Half and half (1 cup)

Milk (2 1/2 cups)

Cheddar cheese (shredded, 2 cups)

Parmesan cheese (shredded, 1/2 cup)