Place potatoes in a large saucepan; cover with water. Bring water to a boil, and cook 15 minutes or until potatoes can be easily pierced with a fork. Add green beans during last minute of cooking; drain and set aside.
While potatoes and beans cook, combine 1 tablespoon olive oil, parsley, mustard, vinegar, and 1/4 teaspoon salt in a large bowl; stir with a whisk. Season to taste with black pepper. When potatoes are cool enough to handle, slice larger pieces in half, and add to bowl. Toss to coat, and set aside.
Cover chicken breasts with parchment paper; pound to an even thickness using a mallet or small, heavy skillet. Sprinkle with remaining 1/4 teaspoon salt and season with black pepper.
Place flour in a shallow dish; dredge chicken in flour. Heat remaining 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add chicken to pan, and cook 68 minutes or until chicken begins to brown; turn to brown other side. Add shallots, and cook 56 minutes or until a meat thermometer inserted into the thickest part of the chicken breast registers 165°. Add lemon juice and tarragon, and turn chicken until evenly coated.
Place chicken breast on each of 4 serving plates; serve with potato salad.
