In a small nonreactive bowl, whisk together the vinegar, cornichons, the 1/4 cup minced dill, the shallot, mustard, sugar and the 1/2 tsp salt until the sugar dissolves. Slowly whisk in the olive oil until well blended to make a vinaigrette. Taste and adjust the seasoning with salt.
In a large saucepan, combine the potatoes and the 1 Tbsp salt, add water to cover the potatoes by 1 inch and bring to a boil over high heat. Reduce the heat to medium, cover partially and simmer until the potatoes are just tender when pierced with a knife, 7 to 9 minutes.
Fill a large bowl two-thirds full with ice water. Drain the potatoes in a colander and immediately transfer them to the ice water. Let stand until cool, then drain again. Cut each potato into quarters and transfer to a large bowl.
Whisk the vinaigrette to recombine, then drizzle it over the potatoes. Add the celery and radishes and stir gently. Taste and adjust the seasoning with salt.
Transfer the salad to a serving bowl and sprinkle with the 2 Tbsp chopped dill. Serve immediately, passing the crème fraîche for dolloping at the table.