For the dressing, whisk together the mayonnaise, crème fraîche, tarragon, lemon juice, olive oil, mustard, salt and pepper in a medium bowl. Set aside.
Bring a medium saucepan of salted water to a boil, add the green beans and simmer 2 to 3 minutes, until just tender but still bright green. Drain and rinse under cold water to cool quickly. Drain well on paper towels.
Steam the potatoes until tender when pierced with the tip of a knife, 15 to 17 minutes. When cool enough to handle (but not fully cooled) peel the potatoes and cut into 1-inch dice.
In a large bowl, combine the potatoes (while they’re still warm) with the dressing, green beans, celery, sweet onion and green onions. Toss gently to mix.
Cover the salad with plastic wrap and refrigerate until ready to serve, at least 30 minutes. Taste the salad again before serving, adding more salt and pepper to taste. If the salad seems a bit dry, stir in a little more olive oil before serving.
The dressing can be made up to 2 days in advance and refrigerated. The salad is at its best made at least 30 minutes before serving so flavors can meld; a day in advance is even better. Serve the salad at room temperature; too cold and the flavors will be dulled. If the salad has been refrigerated, let it set at room temperature for 15 to 20 minutes before serving.