Preheat oven to 200F.
*If you can’t find chicken cutlets, you can butterfly 2 chicken breasts and pound them thin yourself. This is a little bit of next-level cooking. It’s not difficult but it requires a bit of finesse and being careful. To butterfly the chicken breasts, place your hand flat on top of the chicken and cut through the breast horizontally with your knife turned parallel to the cutting board — be careful to keep your fingers out of the way! Place the chicken pieces between two pieces of plastic wrap and pound them to about 1/3 -inch thick using a mallet or small pan. Then cut each cutlet in half so they fit on the buns. . If you are unsure about making the cutlets yourself, many butchers are unbelievably helpful and if you ask them to cut or pound the chicken, 8 out of 10 times they will. I suggest bringing them some granola for Christmas.
Place the chicken cutlet pieces in a single layer on a baking sheet and season them all over with salt and pepper. Set aside while you set up the breading station.
In three separate shallow containers, place the flour, the whisked eggs, and the panko. Season each with salt. Working one at a time, place a piece of chicken in the flour and coat it completely. Next, dip it in the whisked eggs, then remove and allow any extra egg to drip off. Finish by dredging it in the panko,pressing gently to coat completely. When alternating between wet and dry ingredients, I use one hand exclusively for the wet ingredients and the other for the dry ingredients so the flour and panko (and my fingers!) don’t become gloopy.
Place breaded chicken on a parchment-lined baking sheet while you finish the remaining cutlets.
Add all the mayo ingredients to a small bowl and mix until thoroughly combined. Set aside.
To cook the cutlets, add 3 tablespoons of olive oil to a large, non-stick skillet over medium-high heat until it starts to shimmer. Working in batches to avoid overcrowding the pan, add a few breaded chicken pieces to the pan and cook until golden brown, about 3 minutes on each side. Keep an eye on them and flip them the minute they start to brown, turning the heat down to medium if they start to brown too fast. Transfer the finished cutlets to a parchment-lined baking sheet and transfer to the oven to keep warm while you cook the remaining pieces.
Remove any large, breaded pieces from the pan with a slotted spoon or tongs as you go to prevent burning. Add 1 to 2 more tablespoons of olive oil, if needed.
Slightly toast your buns. Slather plenty of the caesar dressing or mayo on the bottom side of the buns and place a piece of crispy chicken on top. Top the chicken with lettuce, a squeeze of lemon and a sprinkle of parmesan. Close the sandwiches and enjoy!