Add the olive oil, vinegar, lemon zest, lemon juice, mustard, shallot, 1/2 teaspoon salt, and 1/2 teaspoon pepper to a glass jar, screw on the lid and shake vigorously to emulsify.
Add the lettuce, chives and radishes to a large serving bowl. Add about half of the dressing and toss to combine. Add the Parmigiano-Reggiano and toss once again. Season with salt a pepper to taste. Serve extra dressing on the side.