To hard-cook the eggs, place the eggs in a saucepan and add water to cover them 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. Set aside to cool completely.

To peel the eggs, once the eggs have cooked and cooled, 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.

 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 place them in the work bowl of a food processor fitted with the metal blade. Blend the yolks, mayonnaise, butter, mustard, and cayenne, and mix until smooth; season with salt and pepper. Add the finely chopped tarragon.

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

To assemble the eggs, when ready to serve, pipe the yolk mixture into the whites. Garnish with additional herbs and serve immediately.

 Making ahead: Unpeeled hard-cooked eggs can be refrigerated for up to 1 week. Or prepare the eggs, but don’t assemble, up to 8 hours in advance of serving; refrigerate the whites covered with a damp towel in an airtight plastic container. Store the egg-yolk mixture in the piping bag with the tip also covered in a damp paper towel. Knead the yolk mixture slightly to soften before filling the yolks. The eggs may also be assembled and stored covered in the refrigerator for up to 2 hours. Any longer and the yolk mixture starts to form a crust.