Greek-style chicken cakes: combine all of the Greek-style chicken cake ingredients in a large bowl EXCEPT for the vegetable oil and panko bread crumbs (they are both used during the cooking process). Mix everything up very well and then divide into 4 or 6 equal portions.
Form each portion into round patties that are at least 3/4 inch thick and about the diameter of the buns you plan to use. Place each patty on a parchment paper-lined sheet pan or plate covered with plastic wrap until it is time to cook. The patties are easier to handle when they are cold, so if you can rest them for 4 hours or more, they will be firmer to the touch. The patties can be rested in the fridge like this for up to 3 or 4 days.
Crispy chicken skin: if you used a rotisserie chicken, make sure to remove as much skin as possible. Place the skin in a flat layer on a sheet pan and bake at 400°F (205°C) for 20 to 25 minutes or until the skin has firmed up and is crispy like bacon. Store in the fridge for up to a week.
Cooking Greek-style chicken cakes: when it is time to make sandwiches, remove the chicken cakes from the fridge and add 1 cup of panko bread crumbs, 1/2 teaspoon of salt, and 1/2 teaspoon of pepper to a large plate or bowl. If you're not cooking all of the chicken cakes at one time, you only need about 2 tablespoons of panko crumbs and large pinches of salt/pepper per chicken cake.
Coat each chicken cake in the panko mixture on both sides.
Add vegetable oil to a medium to large skillet over medium heat. Once the oil starts shimmering, add chicken cakes and cook for around 4 minutes per side. Everything inside the cakes is fully cooked, so we simply need to make sure the middle is warm and the exterior panko bread crumbs have browned and turned crispy. This should take about 3 to 4 minutes per side.
Remove the chicken cakes from the pan and place them on a paper-towel-lined plate if you want to remove some of the exterior oil from frying.
Tomato, red onion, and feta salad: in a small bowl, combine all of the salad ingredients and toss. Taste to see if it needs more salt or pepper and add those as needed. Place aside until it's time to make the sandwich.
Sandwich assembly: toast your buns and then add tzatziki sauce to the bottom of each bun.
Top the sauce with lettuce and then a Greek-style chicken cake. Add a crispy chicken skin on top of the cake and then top with some of the tomato, red onion salad.
Add more tzatziki to the top bun if desired and close the sandwich. Serve and enjoy.