In a large pot over medium heat, melt 1 1/2 Tbsp of the butter. Add carrots, celery and onion and saut? for 3 - 4 minutes. Add garlic and saut? for an additional 30 seconds.

Stir in chicken broth, potatoes and thyme; season with salt and pepper to taste. Bring to a boil over medium-high heat, then reduce heat to medium, cover, and cook for 15 minutes. Stir in broccoli and cook for 5 minutes longer or until vegetables are tender.

Meanwhile, melt the remaining 4 Tbsp butter in a medium saucepan over medium heat. Stir in flour and cook, whisking constantly for 1 minute. While whisking vigorously, slowly pour in milk and whisk until no lumps remain.

Cook, stirring constantly, until mixture begins to thicken. Stir in heavy cream. Remove from heat and, once all vegetables are tender, pour into the soup and stir.

Remove from heat and stir in cheddar and parmesan cheese until melted. Serve warm.