Place the 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, salt and pepper. Add to the egg and stir to combine.
Cut the 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 the sandwiches on a platter, cover with plastic wrap and refrigerate until ready to serve.
Notes
FOR PERFECT HARD BOILED EGGS: Place eggs in a saucepan and cover with water. Bring to a boil. Then turn off the heat, cover the pan and let sit for 15 minutes. Transfer the boiled eggs to a bowl of ice water, then peel and they are ready for eating or using in recipes. Eggs that are one week old work better for hard-boiling than fresher eggs.