Place hard-boiled eggs in a large shallow bowl and mash the eggs with a fork until they are in very small pieces. Add the celery.
In a separate bowl, combine the mayo, avocado, mustard, celery salt, and pepper. Add to the egg and stir to combine.
Cut crusts off the white bread and cut in half twice (4 equal pieces per slice).
Assemble the sandwiches: Place a slice of cucumber on one slice of bread, add a spoonful of egg salad, and top with another slice of bread. Repeat with remaining ingredients.
Place sandwiches on a platter, cover with plastic wrap and refrigerate until ready to serve.