Thinly slice the chicken breasts lengthwise to make thin chicken cutlets. You want THIN slices - ones you can almost see through! I get 3 to 5 thin slices from one chicken breast. 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. I like to heat the oil for a good 5 minutes before adding the chicken. Watch the heat the entire time - if it seems to be too hot, make sure to lower it!
Dip each piece of chicken into the egg, then into the breadcrumbs, pressing gently so the crumbs adhere.
Add the chicken 2 to 3 pieces at a time. Cook on each side until deeply golden, about 3 to 4 minutes per side. This should cook the chicken the entire way through, but take the temperature to make sure it is 165 degrees F inside if you’re unsure. Remove each piece of chicken and place it on the paper towel-lined sheet. Repeat with remaining chicken.
To make the dressing, add the garlic, yogurt, parmesan, mustard, vinegar, anchovy paste, lemon juice, salt and pepper to a food processor or blender and blend until pureed. With the processor still on, stream in the olive oil until a creamy dressing forms. (This may make more dressing than needed, but you can store it sealed in the fridge for about 3 to 4 days and stir well before using.) You can make this ahead of time!
Place the romaine in a large bowl and toss with a big pinch of salt and pepper. Drizzle with a few tablespoons of the caesar dressing. Add in the finely grated and shaved parmesan. Toss the lettuce well so everything is coated.
Assemble the sandwiches by placing a cutlet (or 2) on a bun and topping it high with caesar salad. You can serve the sandwich with extra salad too!