Put the potatoes in a large pot and cover by an inch with cold, salted water. Bring to a boil over high heat, and then reduce to a simmer and cook until fork tender, about 15 minutes.
Place the eggs in a small saucepan and cover with cold water. Bring to a boil, cover and remove from the heat. Allow the eggs sit in the pan for 13 minutes. Run cold water over the eggs to cool. Peel and roughly chop the eggs.
Meanwhile, in the mixer bowl of a stand mixer fitted with the paddle attachment, combine the mayonnaise, mustard, dill relish, shallots, celery, green onions, sweet relish, vinegar, lemon zest, lemon juice, paprika, salt and pepper.
When the potatoes are fork tender, drain in a colander and immediately add to the dressing. Mix on low speed, add the eggs to the potatoes and mix until incorporated. Increase the speed and whip to combine all, leaving the mixture with plenty of lumps in a very creamy dressing. Taste and add more salt, if necessary. Remove from the bowl, spread in a sheet pan and cover with plastic wrap to quickly chill. Chill for at least 2 hours before serving.