Melt 3 tablespoons butter in heavy medium pot over medium-high heat. Add broccoli stems and onion; sauté until onion is translucent, about 6 minutes. Add garlic and tarragon; sauté 1 minute. Add stock; bring to boil. Simmer uncovered until broccoli is tender, about 15 minutes. Stir in cream.
Mix remaining 3 tablespoons butter with flour in small bowl to make paste. Whisk paste into soup. Add broccoli florets. Simmer until soup thickens and florets are tender, stirring frequently, about 5 minutes. (Can be made 1 day ahead. Cool slightly, then refrigerate uncovered until cold. Cover and keep refrigerated. Bring to simmer before continuing.) Preheat broiler. Place 6 ovenproof soup bowls on baking sheet. Divide soup among bowls. Sprinkle 1/3 cup cheese over each. Broil until cheese melts and bubbles around edges, about 4 minutes.