Boil and Peel the Eggs: Place the eggs in a medium saucepan and cover with water about 1 inch above the eggs. Cover the pan and bring the eggs to a full boil. Remove the pan from the heat and let stand for 14 to 16 minutes. Pour off the hot water and cover the eggs with cold water. When cool enough to handle, peel.
Prepare the Filling: Slice 6 of the eggs in half, lengthwise. Turn the yolks into a medium bowl with the remaining unsliced egg. Mash until fairly smooth. Add 3 tablespoons of mayonnaise, mustard, relish, and hot sauce (if using). Mix well, adding more mayo, as needed for moistness. Taste and season with salt and pepper.
Fill the Eggs: Use a teaspoon or small scoop to fill the egg white halves and place them in a plate designed for deviled eggs or nestle them on lettuce or spring greens on a serving plate.
Garnish: Sprinkle with paprika or place pimiento strips on the tops of the eggs.