Boil potatoes: Add cubed potatoes to a large pot with salted water and simmer 15 minutes until fork-tender.
Cook eggs: Prepare hard-boiled eggs using your preferred method, being careful not to overcook the yolks completely.
Prepare eggs: Peel eggs, separate yolks from whites, mash yolks smooth, and dice whites for later use.
Make dressing: Combine mashed yolks, mayo, mustard, butter, vinegar, salt, and pepper in a large mixing bowl.
Mix salad: Gently fold cooled potatoes, diced egg whites, celery, green onions, and relish into creamy dressing.
Chill and serve: Refrigerate for at least 30 minutes, then garnish with paprika and some green onion before serving at your gathering.
