Melt butter in a Dutch oven over medium-high heat. Add chicken; saute until lightly browned, about 3 minutes. Add onion and garlic; saute until softened, about 5 minutes. Stir in mushrooms; saute until tender, 5 to 10 minutes. Pour in chicken stock and 2 tablespoons tarragon; reduce heat to low. Season with salt. Cover and simmer soup until flavors are combined, about 25 minutes.
Stir cream into the soup; cook until heated through but not boiling. Serve soup with pepper and the remaining tarragon on top.
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