Combine mayonnaise, Dijon, vinegar, shallots, tarragon and lemon juice in a large bowl for the dressing; set aside.
Boil potatoes in salted water for 10 minutes. Add green beans; cook until potatoes are tender when pierced, 3-4 minutes. Remove potatoes and beans; plunge the vegetables in ice water to cool. Drain, then quarter potatoes.
Toss potatoes and green beans with dressing to coat; season with salt and pepper. Serve salad chilled or at room temperature.