Cheesy Vegetable Chowder {AKA Broccoli Cheese Potato Soup}

5 1/2 Tbsp butter, divided
1 1/3 cups chopped carrots (about 3)
1 cup chopped celery (2 stalks)
1 cup chopped yellow onion (1 small)
2 cloves garlic, minced
3 cups low-sodium chicken broth
3 1/2 cups peeled and cubed russet potatoes, cut 1/2-inch to 3/4-inch thick (2 large)
3 cups chopped broccoli florets (from about 2 heads)
1/4 tsp dried thyme
Salt and freshly ground black pepper
6 Tbsp all-purpose flour
3 cups milk (preferably 1% or 2%)
1/2 cup heavy cream
2 cups shredded sharp cheddar cheese (8 oz)
1/3 cup finely shredded parmesan cheese (1 oz)