Sprinkle chicken with salt and pepper. In a 6- to 8-quart Dutch oven cook chicken, half at a time, in hot oil over medium-high heat until brown. Drain off fat. Return all of the chicken to Dutch oven.
Press cloves into 1 of the onion halves. Using 100-percent-cotton kitchen string, tie parsley sprigs into a bundle. Add onion halves, parsley sprigs, wine, broth, carrot, celery, and bay leaf to Dutch oven. Bring to boiling, stirring to scrape up any crusty brown bits from bottom of pan. Reduce heat. Simmer, covered, about 30 minutes or until chicken is tender.
Meanwhile, in a covered large saucepan cook haricots verts in boiling salted water for 4 to 6 minutes or until crisp-tender; drain in a colander. In the same saucepan cook and stir mushrooms and shallots in 1 tablespoon butter over medium heat until tender. Return haricots verts to saucepan; cover and set aside.
Strain chicken mixture through a fine-mesh sieve; remove and discard vegetables and herbs. Transfer chicken to a bowl; cover and keep warm. Wipe out Dutch oven. Return strained cooking liquid to Dutch oven. Bring to boiling; reduce heat. Boil gently, uncovered, until reduced to 2 cups.
In a small bowl stir together flour and 3 tablespoons softened butter to make a smooth paste. Gradually whisk paste into reduced cooking liquid. Cook and stir until thickened and bubbly; cook and stir for 1 minute more. Stir in whipping cream; heat through. Stir in chicken, haricots verts mixture, and lemon juice. Cook until heated through, stirring occasionally.
To serve, ladle stew into bowls over rice. Garnish each serving with tarragon.