Preheat the oven to 400°F / 204°F
Slice the hoagie rolls open and spread them out, cut-side up on a large sheet pan. Set aside.
In a mixing bowl, stir together the chicken with 1 Tablespoon cooking oil, garlic powder, Worcestershire sauce, 1/2 teaspoon salt, and pepper.
Heat a large skillet over medium-high heat. Add 2 Tablespoons of oil. When the oil begins to shimmer, add half the chicken and cook on one side without stirring until golden brown. Turn the chicken and continue cooking until cooked through. Put the cooked chicken in a large bowl. Cook the remaining chicken in the same way and add it to the bowl.
Return the skillet to medium-high heat and add the remaining 1 Tablespoon of oil. Add the sliced bell peppers and onions. Season the vegetables with 1/2 teaspoon salt. Cook, stirring occasionally, until the bell peppers and onions are very tender, 10 to 12 minutes. (If the pan looks dry at any point, add a splash of water.)
Add the peppers and onions to the chicken and stir everything together.
Spread the insides of the hoagie rolls with softened butter. Add cheese slices to both sides of the rolls. Fill the rolls with chicken, peppers, and onions.
Bake the sandwiches in the oven until the cheese is melted and bubbly, 8 to 10 minutes.
Serve warm.