Boil the potatoes in a large pot of salted water. The potatoes are ready when a knife slides in easily, but the very center should still feel just a touch firm. This could take between 8 to 15 minutes depending on the size of the potatoes. If your potatoes vary in size or type, fish the individual pieces out of the water to let cool as soon as they’re ready.
Drain the potatoes and transfer them to a medium-size bowl, along with the scallions, then immediately drizzle the pickle brine over the potatoes. Let the potatoes cool slightly. As they cool, toss them in the bowl a couple of times to help distribute the pickle brine and encourage it to soak in.
In a small bowl, whisk together the mayonnaise, mustard, horseradish, and dill, along with a healthy pinch of salt and a few grinds of black pepper.
Once the potatoes have cooled, toss them with the dressing and adjust seasonings to taste.
Serve immediately or let the potato salad hang out, covered, in the fridge overnight to let the flavors meld even more. It will keep for 3 to 5 days in the refrigerator.