Saute the onions and garlic in butter in a medium skillet over medium heat, stirring constantly until tender. Set aside.
Bring the chicken broth and green beans to a boil in a large saucepan. After about 15 min., add potatoes. Cook until tender. Add the reserved onion and garlic mixture. Stir well.
Transfer mixture into blender jar; cover and process until smooth. Repeat with remaining mixture, if necessary. Return pureed mixture to the saucepan and bring to a boil. Stir in cream, and lemon juice and tarragon. Cook until thoroughly heated. Garnish with a dollop of cream and freshly ground black pepper.