Bring a large pot of water to a boil over high heat, add the potatoes, and cook until tender when pierced with a fork, about 20 minutes. Remove from the water with a slotted spoon or strainer; set potatoes aside but maintain the water's boil.
Meanwhile, spray the broiler rack with nonstick cooking spray; preheat the broiler. Broil the chicken breasts 5 inches from the heat, turning occasionally, until cooked through, about 10 minutes. Set aside.
Add the green beans to the boiling water; blanch until crisp-tender (no more than 2 minutes), then drain in a colander set in the sink.
When the chicken is cool enough to handle, cut it into bite-size chunks. Slice the potatoes into 1-inch chunks, cut the beans into 1-inch strips, and gently toss with the chicken. Add the celery and grapes.
Whisk the lemon juice, broth, and mustard in a small bowl; whisk in the tarragon, salt, and pepper. Drizzle in the olive oil in a slow stream, whisking all the while, until the dressing turns creamy, about 1 minute. Toss the salad with this dressing, taking care not to break up the potatoes. If desired, cover and refrigerate for up to 2 days.