6 tablespoons (3/4 stick) butter, room temperature
2 pounds fresh broccoli, stems and florets separated and chopped into bite-size pieces
1 large onion, chopped
2 garlic cloves, minced
6 1/2 cups chicken stock or canned low-salt chicken broth
1 cup whipping cream
3 tablespoons all purpose flour
2 cups (packed) grated extra-sharp cheddar cheese (about 8 ounces)
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.
Soup before cheese: