Heat your griddle (or skillet) on medium heat. *For griddle cooking, aim for a griddle temperature of about 350-400℉.
Add the sliced raw chicken breast to a large bowl with 2 tablespoons cooking oil and about 2 tablespoons of all-purpose seasoning. Stir to combine and set aside.
Add about 1 tablespoon of cooking oil to the griddle and spread it around with your spatula. Add the sliced bell peppers and onions to the oiled griddle. Season vegetables with all-purpose seasoning or salt and pepper, and sauté for 2-3 minutes, stirring occasionally.
While vegetables are cooking, cut the bread for sandwiches about ¾ of the way through so that you can open it up, similar to a sub sandwich. Set aside.
When the vegetables are almost done cooking, add the sliced chicken breast to a clear area on the griddle, spreading the pieces out into a thin even layer. Sauté until almost cooked through, about 2-3 minutes depending on the griddle temperature. Then combine the chicken and vegetables together. Sauté the chicken and veggie mixture just until chicken is cooked through.
On the griddle, portion out chicken and veggie mixture into three long "piles" about the same length of your sandwich rolls. Top each meat and vegetable portion with 2 slices of cheese. Add some of the chopped sweet cherry peppers on top, if using. Allow the cheese to melt for about 1 minute, then mix each pile together to incorporate the cheese slices.
Add 1-2 additional slices of cheese on top of each pile. Then place a hoagie roll opened up and cut-side down directly on top of each chicken pile. Let steam for about 1 minute.
Use a griddle spatula to carefully transfer the chicken cheesesteak mixture to the hoagie rolls. Wrap in deli paper if desired, then cut in half and serve immediately while hot.