To prepare chicken, place each chicken breast half between 2 sheets of heavy-duty plastic wrap; pound each piece to 1/2-inch thickness using a meat mallet or rolling pin. Combine chicken and buttermilk in a large zip-top plastic bag; seal. Marinate in refrigerator for 2 hours, turning bag occasionally. Remove chicken from bag; discard marinade.
Pat chicken dry with paper towels; sprinkle with 1/2 teaspoon salt. Combine flour, tarragon, and 1/2 teaspoon black pepper in a shallow bowl. Dredge chicken in flour mixture; shake off excess. Heat oil in a large nonstick skillet over medium-high heat. Add chicken; cook 3 minutes on each side or until done.
To prepare salad, place haricots verts into a large saucepan of boiling water; cook 3 minutes. Drain and plunge haricots verts into ice water; drain. Place haricots verts, spinach, arugula, watercress, parsley, and chives in a large bowl. Combine shallots, juice, oil, vinegar, mustard, 1/4 teaspoon salt, and 1/8 teaspoon pepper in a small bowl, stirring with a whisk. Spoon 3 tablespoons shallot mixture over spinach mixture; toss gently to coat. Place 2 cups salad mixture onto each of 4 plates; top each serving with 1 chicken breast half. Drizzle chicken evenly with remaining shallot mixture.