Add the potatoes to a pot and cover with water, plus an extra inch. Add a tablespoon of kosher salt and cover. Bring the potatoes to a boil over medium heat and allow them to boil until just fork tender. Drain them into a colander.
Place the potatoes back into the hot pot and pour 1/4 cup pickle brine over them. Allow them to cool completely.
Add half of the mayonnaise and the remaining 1/4 cup of pickle brine to a bowl and stir to combine and loosen up the mayo. Add the remaining ingredients for the dressing to the bowl and stir to completely combine.
Add the cooled potatoes, eggs, celery, both onions, and pickles to a large bowl. Pour the dressing over the top. Use a rubber spatula or spoon to gently fold the mixture until completely coated and combined.
Cover the potato salad and let it refrigerate for 1-2 hours.