Wash but do not peel the cucumber. Cut the cucumber in half down the center. Slice the one half into thin slices and then into half moons.
Grate the other half of the cucumber into a bowl using a box cheese grater on the thicker shredded side. Add one teaspoon of celery salt to the grated cucumber and toss to coat. Let this sit.
In a bowl, or a bowl of a stand mixer, add the whipped cream cheese. Thinly slice half the chives and add to the cream cheese along with a few pinches of celery salt. Beat with a hand mixer or stand mixer on low for a minute.
Add the grated cucumber to the cream cheese and mix on medium to high for 2 to 3 minutes.
Sprinkle the sliced cucumbers with a little bit of the celery salt and set aside. I usually put these in a bowl.
Run your knife through the shaved ham to make even smaller bits of ham.
Spread your bread slices out on a platter or baking sheet.
Spread the cream cheese mixture on each slice. I use a little less than a ½ tablespoon on each one. I make this easier on myself by adding my cream cheese mixture to a pastry bag or makeshift pastry bag made out of a Ziploc bag. *pictured in post. I squeeze a dollop on each bread square.
Add ham to each bread slice, pressing it slightly into the cream cheese.
Add a cucumber slice to each bread slice and secure it by pushing it into the cream cheese.
Thinly slice the rest of the chives or cut with kitchen shears and sprinkle over the top of all the sandwiches. Then sprinkle them all with dried dill.
Cover and refrigerate up to six hours before serving.
