Place the potatoes in a large pot and cover with 1 inch of cold water. Add the salt and bring to a boil over high heat, then reduce to a simmer and cook until a paring knife slides easily into the center of the largest chunk, about 10 to 12 minutes.

Whisk the vinegar, mustard, and oil together in a large bowl until the mixture is thick and emulsified.

Drain the potatoes into a colander and let them sit for 2 minutes—the surface of the potatoes should look matte as excess moisture evaporates.

Transfer the warm potatoes to the bowl with the vinaigrette and add the red onion. Fold gently with a rubber spatula so the potatoes stay intact while soaking up the liquid.

Add the radishes, capers, and cornichons. Toss once more to distribute the components evenly and season with salt and pepper to taste.

Serve warm or at room temperature, while the radishes are still crisp.