Place chicken in food processor and pulse until very finely chopped, transfer to bowl. For crab, pick through meat for shells and mash with fork into small bits.
To processor bowl, add celery, onion, garlic, chili pepper, thyme and parsley, and very finely chop by pulsing; add to chicken or crab. To the bowl, add mustard, hot sauce, relish, Worcestershire, and, if making with crab, lemon juice; mix to combine and season with salt and pepper to taste. Transfer to serving bowls; chicken is garnished with scallions and crab is garnished with chopped toasted almonds and chives. Surround spread with toast points, chips and veggies.