Thinly slice the chicken breasts lengthwise to make thin chicken cutlets. Season the chicken all over with salt and pepper. Line a baking sheet with paper towels.
Place the eggs in a shallow plate or bowl. Stir together the breadcrumbs and parmesan in another plate or bowl.
Heat 1 to 2 tablespoons of olive oil in a large skillet over medium heat. Watch the heat the entire time.
Dip each piece of chicken into the egg, then into the breadcrumbs, pressing gently so the crumbs adhere.
Cook chicken 2 to 3 pieces at a time on each side until deeply golden, about 3 to 4 minutes per side. Ensure internal temperature is 165 degrees F.
To make the dressing, add garlic, yogurt, parmesan, mustard, vinegar, anchovy paste, lemon juice, salt, and pepper to a food processor or blender and blend until pureed.
Place the romaine in a large bowl, toss with salt and pepper, and drizzle with caesar dressing. Toss well to coat.
Assemble the sandwiches by placing a cutlet (or 2) on a bun, topping with caesar salad.