Chop up a chunk of the Parmigiano-Reggiano, reserving a 2-inch piece for garnish. Using a mini food processor or high-power blender, grate the Parmigiano-Reggiano into a fine powder.
For the dressing, smash the garlic, adding salt to the garlic to break it down and turn it into a paste.
In a bowl placed on a kitchen towel to stop it from sliding, combine the garlic, anchovy paste, egg yolks, 1/2 teaspoon of the Dijon mustard and lemon juice. Taste and season with a little salt, as needed.
Starting with adding a drop at a time, and increasing to a delicate stream, slowly whisk in 1/2 cup of the oil until a creamy, emulsified dressing forms. Add in 3 tablespoons of the grated Parmigiano-Reggiano and adjust the seasoning to taste, adding more salt if needed and definitely adding in some freshly cracked pepper.
Using a large zip-top bag or plastic wrap and a mallet or rolling pin, pound the chicken breasts to 1/4-inch thick pieces. Season generously with salt and pepper.
Using three plates or shallow bowls, make your dredging station. In one bowl, combine the flour and garlic powder, in another bowl combine the egg whites, the remaining 1 tablespoon Dijon mustard and 1 tablespoon of water and in the final bowl, combine the plain breadcrumbs, panko, remaining Parmigiano-Reggiano and lemon zest.
Working in an assembly line, dip the chicken breasts into the flour, then the egg wash, then the breadcrumbs. Place on a baking sheet.
Add the remaining 2 tablespoons oil to a large skillet over a medium-high heat. Once the oil is shiny, add 2 of the cutlets to the pan. Cook the chicken until golden brown and cooked through, 2 to 3 minutes per side. Repeat with the remaining cutlets.
Place several spoonfuls of the dressing into a fresh bowl. Using clean hands, add the lettuce, season with salt and toss the salad, making sure the dressing gets into all the nooks and crannies of the lettuce.
Brush the chicken cutlets with the remaining dressing, top with salad and finish with shards of Parmigiano-Reggiano and more freshly cracked pepper. Enjoy!