Scrub, dice potatoes. Place potatoes into a large pot. Fill with water and pour water out and refill with water to 1" above potatoes. Place pot on burner over high heat. Season with 1/2 TBS salt. Bring to a boil, lower heat to a medium and let simmer until potatoes are fork tender.
Drain potatoes. Run cold water over potatoes until they are cooled. Leave potatoes in strainer to air dry.
While potatoes are cooking, you can start the eggs. Hard boil your eggs in the method that do this step. Remove eggs and place into a boil of cold icy water to stop the cooking process and to also allow eggs to cool for easy handling.
Peel eggs and cut in half lengthwise. Remove yolks from hard boiled eggs and place yolks into a small bowl.
Add in mayonnaise, 1/8 tsp salt, 1/8 tsp black pepper, and mustard into same bowl with yolks. Using a fork, mix ingredients together making sure to break up yolks.
Divide yolk mixture between the 12 egg white halves. (These should look like deviled eggs.) Cut the deviled eggs into quarters and place into a large mixing bowl.
To the mixing bowl with the deviled eggs pieces, add in the cooked, drained, and cooled potatoes, diced celery, minced shallots, relish, and 1/4 tsp smoked paprika. Mix well to combine.
Place in a serving bowl and sprinkle with paprika.