In a large pot, add the potatoes and cover with water. Season with a few generous pinches of salt. Bring to a boil, then reduce the heat to moderately-low and simmer until the potatoes are tender, 15 to 20 minutes depending on the size. Drain well and let cool completely. Halve the potatoes, or quarter if large, and transfer to a large bowl.
Meanwhile, prepare an ice bath. In a medium pot of boiling water, cook the eggs, stirring occasionally, until the whites are set and the yolks are still soft, about 7 minutes. Transfer to the prepared ice bath and let cool for 5 minutes. Drain and peel the eggs.
In a medium bowl, whisk the mayonnaise with both mustards, lemon zest, lemon juice and pickle brine. Season with salt and pepper.
To the bowl with the potatoes, add the chopped celery and red onion. Pour half of the dressing over the potatoes and gently toss to coat. Add more dressing as desired. Season with salt and pepper. Transfer the potato salad to a serving bowl or platter.
Using a paring knife, cut the soft boiled eggs into quarters and arrange on top of the potato salad. Sprinkle with the celery leaves, dill and chives. Garnish with chopped pickles, crushed potato chips and more black pepper. Serve right away.