To make hard-cooked eggs: Place eggs in single layer in saucepan; cover with water until about 1 inch above eggs. Bring to a boil, uncovered. Then remove from heat, cover pan and let sit for 15 minutes. Immerse in cold water until eggs are no longer warm.
To make the dressing: Combine 3 egg yolks, mayonnaise, yogurt, buttermilk, mustard, cayenne and black peppers, and salt in a blender or food processor; purée until smooth. Spoon into a bowl, then stir in the chives and pickle relish.
Finely grate 3 egg whites; add them to the dressing. The yield is 1 1/4 cups. Cover and refrigerate.
To make the salad: Line a small baking sheet with a double thickness of paper towels. Core and rinse the lettuce, then cut it into 4 wedges, placing the wedges on the baking sheet with a cut side down. Take enough paper towels to cover the wedges with a double thickness. Blot the tops of the wedges with the paper towels and cover the wedges with them. Refrigerate while you assemble the salad toppings.
Cut 1/4-inch of skin and flesh lengthwise from all sides of the cucumber. Cut each of those slices into lengthwise strips 1/4-inch-thick. Line up the strips and cut them widthwise all at the same time into 1/4-inch cubes.