Prep your vegetables. Give the celery a rough chop into manageable pieces — they don't need to be tiny, the blender will handle that. Do the same with the green onions, trimming the ends and cutting them into a few pieces. A rough chop is all you need here.
Add everything to the blender. Place the celery, green onions, roughly cubed ham, chopped dill pickles, and the boiled egg (cut into a few pieces) into your blender or food processor. A food processor actually works a little better here — it gives you more control over the texture. But a blender works fine; you'll just need to stop and scrape down the sides once or twice.
Pulse — don't blend. This is the key step. You want to pulse in short bursts, not run it continuously. We're going for a chunky, slightly coarse texture — some nice pieces of ham and bits of pickle and onion throughout. If you run it too long, it'll turn into a paste, and we don't want that. Give it a few pulses, check the consistency, and keep going until it looks right to you.
Make the dressing. In a separate bowl, combine the mayonnaise, Dijon mustard, pickle juice, and black pepper. Give it a good stir until it's smooth. Taste it before adding any salt — the pickle juice brings some saltiness on its own, and the ham will too.
Combine and taste. Add the dressing to the ham mixture and stir everything together until it's well coated. Now taste it. The ham and pickle juice should provide most of the salt you need. Add just a little if you feel it needs it, but go easy — you can always add more, but you can't take it back.