Place potatoes in a large saucepan; cover with water. Bring water to a boil and cook potatoes 15 minutes or until potato is easily pierced with a fork. Add green beans during the last minute of cooking; drain and set aside.
While potatoes 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 or fourths and add to bowl. Toss to coat and set aside.
Cover chicken breasts with parchment paper; pound to an even thickness using a meat 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. Warm remaining 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add chicken to pan, and cook 6 to 8 minutes until chicken begins to brown; turn to brown other side. Add shallots, and cook 5 to 6 minutes or until a meat thermometer in the thickest part of the chicken breast registers 160ï¿½. Add lemon juice and tarragon, and turn chicken until evenly coated.
Transfer chicken to a serving platter and scrape any remaining shallots into potato salad. Let stand 5 minutes before serving.
The chicken breast was just OK, but the potato salad was great. The only thing I did differently was cut the potatoes before I boiled them because I'm too impatient to wait for them to cool and cut them.
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