In a large pot melt butter. Add carrots, celery and onion and saute 3-4 minutes, Add garlic and saute 30 seconds longer.
Stir in chicken broth and potatoes and season with salt and pepper to taste. Bring to a boil, then reduce to medium low heat, cover with lid and cook 15 minutes. Add broccoli and cook for 15 more minutes.
Carefully add half the soup into blender using a ladle, slowly add the heavy cream to the blender.
Add puree back into the pot.
Continue stirring mixture until it comes to a boil and thickens.
Remove from heat and stir in cheddar and Parmesan cheese until melted. Serve warm.