Preheat the oven to broil. Line a baking sheet with parchment paper or heavy-duty aluminum foil.
Carefully use a sharp kitchen knife to slice a V to remove the top and center of the deli rolls. Be very careful not to cut the bottom of the roll.
Divide the 2½ tablespoons of the butter between the 4 deli rolls. Spread the butter on the inside of the deli rolls.
Place the buttered rolls onto the prepared baking sheet. Place the rolls in the oven to toast for 2-3 minutes or until the bread is golden and toasted to your liking. Remove the rolls from the oven.
Add the sliced chicken breast to a medium-sized mixing bowl. Sprinkle the dried Italian seasoning, kosher salt, and cracked black pepper over the chicken. Toss to completely coat.
Add 1 tablespoon of extra virgin olive oil to a skillet over medium-high heat. Add the seasoned chicken to the heated skillet. Cook for 4 to 5 minutes, turning the chicken strips halfway through, until cooked through and no longer pink. Transfer the cooked chicken to a heat-safe bowl.
Return the skillet to the heat. Add the remaining extra virgin olive oil and the remaining butter. Once melted, add the sliced green bell peppers and sweet yellow onion. Sauté for 2 to 3 minutes.
Add the sliced mushrooms and minced garlic. Sauté for another 2 to 3 minutes until the onion is opaque and the mushrooms are cooked through.
Add the cooked chicken back into the skillet and stir to combine.
Divide the chicken and veggie mixture between the toasted deli rolls. Stagger the layers of chicken, cheese, and the veggie mixture.
Place the sandwiches back under the broiler for just 1 to 2 minutes or until the cheese is melted.
