Heat 1 1/2 tablespoons of butter in a large saucepan over medium heat until it melts. Saute the onion, celery, and carrots for three to four minutes. Sauté for another 30 seconds after adding garlic.
Toss in the potatoes, thyme, chicken broth, and salt and pepper, then stir to combine. Reduce heat to medium-low, cover, and simmer for 15 minutes after coming to a boil over medium-high heat. After 5 minutes, stir in the broccoli and continue cooking until the vegetables are soft.
A medium saucepan set over medium heat should be used to melt the remaining 4 tablespoons of butter. Add the flour and whisk continuously while cooking for 1 minute. Gradually add the milk while whisking constantly (whisk well to remove lumps).
After the liquid starts to thicken, whisk in the heavy cream and continue cooking while stirring continually. Once the vegetables are cooked, remove from heat and toss them into the soup. When the cheeses have melted, remove from the fire and toss in the parmesan and cheddar. Heat and serve.