My Favorite Broccoli Cheese Soup

6 Tbsp butter, diced in cubes
1 cup diced yellow onion (about 1/2 large onion)
1 large clove garlic, finely minced
1/4 tsp dried thyme
1/4 cup + 3 Tbsp all-purpose flour
3 cups milk (I used 2%)
2 cups chicken broth
1/2 cup heavy cream
3 cups finely chopped, fresh broccoli pieces*
8 oz. freshly shredded Sharp Cheddar cheese
1 oz. freshly, finely shredded Parmesan cheese (or more to taste)
salt and freshly ground black pepper to taste (I used about 1/2 tsp of each, but this will vary depending on the brand of cheeses and your salt/pepper preference).