For the egg salad: Start by hard boiling your eggs. Put your eggs in cold water in a medium saucepan, place on the stove, and bring to a boil. Turn the stove off, cover the saucepan with a lid, and after 15 minutes run them under cold water. Peel the shells from the eggs (reserve 2 whole eggs for Round 2 Recipe Nicoise Salad with Lemon Garlic Vinaigrette) and chop.
In another bowl, stir together the mayonnaise, paprika, dill, and salt and pepper. Add the chopped eggs and gently toss.
For the ham salad: In a medium bowl, whisk together the mayonnaise, mustard, pickle relish, onion, salt, pepper, and parsley. Add the celery and half the ham (reserve remaining half for Online Round 2 Recipe Ham and Cheese Breakfast Burrito) and toss to coat. Set aside.
For the sandwiches: Cut off the crusts from the wheat and white bread. On a clean surface, place 1 slice white bread down and top with half the egg salad. Top that with 1 slice wheat bread and spread half the ham salad on top. Top with 1 slice white bread. Slice crosswise into triangles. Repeat, alternating the white and wheat bread as the beginning slice, until you have made 4 sandwiches.