Step 1

To hard-cook the eggs, in a large saucepan, cover the eggs with water by 1 inch. Bring to a boil over high heat (you will see bubbles around the sides of the pot). Remove from the heat, cover, and let stand for 12 minutes. Drain the eggs and rinse them under cold running water. Let cool completely.

Step 2

To peel the eggs, remove the shells by tapping each egg gently on the counter or sink all over to crackle it. Roll an egg between your hands to loosen the shell. Peel, starting at the large end, while holding the egg under running cold water; this facilitates peeling and also removes any stray shell fragments.

Step 3

To prepare the filling, halve the peeled eggs lengthwise. Carefully remove the yolks. Set the whites aside. Pass the yolks through a fine-mesh strainer into a bowl or in the bowl of a food processor fitted with the metal blade. Add the yolks, mayonnaise, butter, mustard, and cayenne and stir or blend until smooth; season with the salt and pepper. Stir in the tarragon.

Step 4

Transfer the mixture to a piping bag fitted with a large star tip, or use a medium sealable plastic bag with one of the corner tips snipped off.

Step 5

To assemble the eggs, pipe the yolk mixture into the whites. Garnish with more tarragon and serve immediately.