Peel eggs and cut them in half the long way, and carefully remove the yolks.
Mash the yolks with salt, mayo, mustard and cayenne. Taste and adjust seasoning. Spoon filling back into the whites (or pipe the filling in with a pastry bag to make them look especially nice). Garnish and serve with paprika or parsley, or cover and chill, well wrapped, for up to a day before serving.
Anchovied Eggs: Substitute 1 tablespoon olive oil for the mayo; mash 2 or more minced anchovies with the yolks. Garnish with a piece of anchovy fillet and a couple of capers.
Herb-Stuffed Eggs: Substitute 1 tablespoon olive oil for the mayo. Add 1/4 teaspoon finely minced garlic, 2 tablespoons minced fresh parsley leaves, 1 tablespoon drained and chopped capers (optional) and 1 teaspoon minced fresh tarragon leaves or 1 tablespoon minced fresh basil leaves. Garnish with parsley sprigs or small basil leaves.
Eggs Stuffed With Spinach: This makes enough stuffing for 8 whole eggs. Steam or parboil 6 to 8 ounces of trimmed spinach; drain, squeeze dry, and chop finely. Mash it with the egg yolks, 1/2 cup freshly grated Parmesan cheese, 2 tablespoons butter, a tiny grating of nutmeg, salt and pepper. Taste and adjust seasoning before stuffing the whites.