Chop the cooked ham into 1-inch pieces and prepare the vegetables by roughly chopping the green onions, celery, and hard-boiled egg.

In a food processor, combine the chopped ham, celery, pickles, and green onions. Pulse until finely chopped, maintaining some texture.

Transfer the mixture to a mixing bowl and mix in the mayonnaise, Dijon mustard, and black pepper. Add hot sauce if desired.

Taste and adjust seasoning with kosher salt as needed, then serve on crackers, lettuce wraps, or as a sandwich filling.