Peel potatoes. Slice lengthwise into 1-inch-thick planks. Lay flat and slice again into 1-inch-thick strips. Stack strips together and cut crosswise into even cubes.

In a large pot, cover potatoes with at least 2 inches of water and season generously with salt. Bring to a boil, then reduce heat to medium-high and continue boiling until potatoes are tender and easily pierced with the tip of a knife, 8 to 9 minutes. Drain and transfer to a bowl. Toss with 2 tablespoons vinegar; season with salt and pepper. Let cool completely, about 30 minutes.

Meanwhile, whisk together mayonnaise, mustard, remaining 1 tablespoon vinegar. Stir in celery, cornichons, and scallions. Gently fold mayonnaise mixture into cooked potatoes; season with salt and pepper. Serve, or refrigerate in an airtight container up to 1 day (bring to room temperature before serving, about 30 minutes).