Bring a medium heavy-bottomed saucepan of generously salted water to a boil. Fill a large bowl halfway with ice water.

Carefully add the chicken eggs to the boiling water and cook until hard-cooked, about 11 minutes. Use a slotted spoon to transfer them to the ice bath to cool. Carefully add the quail eggs to the boiling water and cook for 1-1/2 minutes. Remove to the ice bath to chill.

Once cooled, carefully peel the eggs and cut each in half lengthwise. Remove the yolks from the egg halves and transfer them to a food processor fitted with a steel blade, reserving the whites of the quail eggs and discarding the whites of the chicken eggs (or saving them for another use). Add the mayonnaise, mustard, and lemon zest, and process the mixture to a fine paste. Season carefully to taste with salt, bearing in mind that the quail eggs will be topped with caviar.

Transfer the deviled yolks to a pastry bag fitted with a small star tip and pipe the filling into the quail egg white halves. Top each with a scant amount of caviar and transfer to a platter before serving.

Make Ahead: The eggs can be made up to 1 hour ahead of time, covered loosely with plastic wrap, and refrigerated. Serve cold.

Make Ahead: The eggs can be made up to 1 hour ahead of time, covered loosely with plastic wrap, and refrigerated. Serve cold.