Prepare the Chicken: Thinly slice the chicken breasts lengthwise to create very thin cutlets. Aim for slices so thin you can almost see through them; you should get 3 to 5 slices per breast. Season all over with salt and pepper.

Set Up Breading Station: Place the beaten eggs in one shallow bowl. In a second bowl, stir together the seasoned breadcrumbs and 1/3 cup of grated parmesan cheese.

Fry the Cutlets: Heat 1 to 2 tablespoons of olive oil in a large skillet over medium heat. Dip each chicken slice into the egg, then into the breadcrumb mixture, pressing firmly. Add to the skillet and cook for 3 to 4 minutes per side until deeply golden and cooked through. Drain on a paper towel-lined sheet.

Make the Dressing: In a food processor or blender, combine the garlic, yogurt, parmesan, mustard, vinegar, anchovy paste, lemon juice, salt, and pepper. Blend until smooth. While the processor is running, slowly stream in the olive oil until the dressing becomes creamy.

Toss the Salad: Place the chopped romaine in a large bowl. Drizzle with a few tablespoons of the dressing and add the finely grated and shaved parmesan cheese. Toss well until every leaf is coated.

Assemble: Place one or two crispy chicken cutlets on the bottom of a brioche bun. Pile a generous amount of the Caesar salad on top, add the top bun, and serve immediately.