Cut and cover the potatoes. Dice the potatoes into small bite-size pieces, keeping them roughly the same size so they cook at the same rate. Leave the skins on or peel them, whichever you prefer. Put them in a large pot with enough water to cover them by 1 inch, and don’t salt the water.

Simmer until tender. Bring the pot to a boil over medium-high heat, then turn the heat down and let the potatoes simmer for about 8 to 10 minutes from the moment they started boiling. They’re ready when a knife tip slides into a piece with only light resistance.

Hard-boil the eggs alongside. While the potatoes cook, boil the eggs in a separate pot, timing 10 minutes from the point they start to boil. Run cold water over them for a few minutes to cool them, then peel once they’re cool enough to handle.

Mix and chill the dressing. Combine the mayo, celery salt, Worcestershire, Dijon, white vinegar, pickle juice, parsley, and pepper in a small bowl and stir until completely smooth with no streaks. Put it in the fridge until you need it.

Drain and season with vinegar. Drain the cooked potatoes thoroughly and tip them back into the pot they cooked in. Drizzle over the 1 tablespoon of white vinegar and toss gently with two spoons, lifting rather than stirring so the pieces stay whole.

Cool for 20 minutes. Leave the potatoes to cool for 20 minutes, tossing them gently a couple of times to get air circulating and speed things up. They should still be warm at the end, just not hot enough to thin the dressing.

Chop everything else. While the potatoes cool, chop the pickles, celery, onions, and the peeled eggs. Keeping the onion small matters more than the others, since large pieces come through sharply.

Toss and finish. Add the warm potatoes to a large salad bowl with the chopped ingredients and the dressing, then toss until everything is coated. Sprinkle paprika over the top if you like. Eat it right away while it’s still warm, or cover the bowl tightly and chill it for an hour or two so the flavours settle together.