Combine the cayenne pepper, salt, cumin, oregano, pepper, and paprika for the dry rub. Cut each chicken breast 2-3 times lengthwise for thinner slices and coat them in the dry rub.
Cut yellow, green, and red bell peppers into strips and slice the red onions. Transfer the veggies to a frying pan with some olive oil and stir-fry until tender.
Line the inside of a 7-inch springform pan with plastic wrap and cover the bottom of the pan with a layer of seasoned chicken breast. Add a layer of stir-fried veggies on top of the chicken. Sprinkle shredded cheddar cheese on top of the vegetables. Repeat the layering process with the remaining ingredients using the same 3 layers: seasoned chicken breast, stir-fried veggies, and shredded cheddar cheese, with the final layer being seasoned chicken breast. Fold the plastic wrap up over the top layer and freeze the pan for 2 hours.
Preheat the oven to 350°F and heat a pot of cooking oil on the stove. Take the springform pan out of the freezer and remove the plastic wrap. Make sure that the outside of the chicken fajita cake is frozen and firm enough to be breaded. Whisk about 10 eggs in a deep pan and season with salt and pepper. Set up the rest of your breading station with separate deep pans for the flour and bread crumbs. Roll the frozen chicken fajita cake in flour, dredge in egg, then coat in bread crumbs.
Fry the breaded chicken fajita cake in hot oil until crispy, about 10 minutes on each side. Put the fried fajita cake back in the springform pan, transfer it to the oven, and bake at 350°F for 60 minutes.