Place cubed potatoes in a large pot and cover with cold water by 2 inches. Add 1 tablespoon kosher salt. Bring to a boil over high heat, then reduce to a gentle simmer. Cook until potatoes are tender when pierced with a fork, 12 to 15 minutes. Do not overcook - they should hold their shape.

Drain potatoes in a colander and spread on a baking sheet in a single layer. Let cool to room temperature, about 30 minutes. This prevents the dressing from breaking and keeps the potatoes firm.

In a small bowl, whisk together mayonnaise, sour cream, Dijon mustard, pickle brine, and black pepper until smooth. The brine adds tang and thins the dressing just enough to coat evenly.

Transfer cooled potatoes to a large mixing bowl. Add diced pickles, celery, red onion, and chopped dill. Pour dressing over the top and fold gently with a rubber spatula until everything is evenly coated. Taste and adjust salt if needed - the pickles and brine already add saltiness.

Cover and refrigerate for at least 1 hour, preferably 2 to 4 hours, to let flavors meld. Garnish with additional fresh dill just before serving.