Place the potatoes in a large pot and cover with cold water by about an inch. Stir in the salt, bring to a boil, then reduce to a simmer and cook for 15 to 20 minutes, until a fork slides into a potato with a little resistance but the potato still holds its shape. Don't let them go soft, since they'll break apart once you fold in the dressing.

Drain the potatoes and let them sit in the colander for a few minutes so they stop steaming. Meanwhile, whisk together the vegan mayonnaise, mustard, and white wine vinegar in a large bowl until completely smooth.

While the potatoes are still warm, add them to the bowl along with the celery, red onion, dill pickles, pickle brine, and fresh dill. Fold everything together gently with a spatula until every potato is coated, being careful not to mash them.

Season with the black pepper and taste, then add more salt if it needs it. The salad should taste slightly brighter than you want the final version to be, because the potatoes will absorb flavor as the salad chills.

Cover the bowl and refrigerate for at least 1 hour, or overnight for the best flavor. Before serving, stir in another spoonful of vegan mayonnaise if it looks dry, then top with a little extra fresh dill.