Heat 2 tablespoons of olive oil in a medium to large skillet over medium-high heat. Add the chicken breast and season with salt and pepper. Cook thoroughly, about 5-7 minutes.
Meanwhile, heat the other 2 tablespoons of olive oil in a large pot or Dutch Oven over medium heat. Add the red bell pepper, onion, and celery and season with a small amount of salt and pepper (you will be adding salt a few times so be careful not to over salt). Cook until soft, about 5 minutes.
Add the eggplant and season with a small amount of salt and pepper. Cook until the eggplant has a nice sear on it, about 7-8 minutes.
Next, add in the garlic and cook for 1 more minute, the add in the tomato paste. You want to be sure to cook the paste so stir it in with the vegetables and allow it to cook for about 2-3 minutes.
Add in the tomatoes, red wine vinegar, capers, oregano, basil, and sugar and cook 2 minutes. Taste and make any adjustments to the salt and pepper.
Now add in the chicken and the juices from the plate. Stir in the raisins and remove from the heat.
Slice the ciabatta rolls in half. Pull out good amount of bread from inside the top. Butter the outside of the top of the roll.
Scoop about 1 cup of the Chicken Caponata onto the bottom of the bun, add a slice of cheese, and place the top on the roll.
Turn the roll over and butter the bottom while holding the sandwich in your hand.
Place the sandwich on the press or grill and close the lid. Allow to grill for about 3 or 4 minutes.
Alternatively, before assembling the sandwich, you can heat a cast iron skillet on the stove with 1 tablespoon of olive oil. Aslo heat another skillet with 1 tablespoon of butter.
Assemble the sandwich as above, only don't butter the bottom of the roll.
Place the assembled and buttered sandwich in the pan with the butter. Place piece of aluminum foil between the cast iron skillet and the sandwich and place the cast iron skillet on top of the sandwich.
Cook for 3-5 minutes. Once you remove the cast iron skillet, take a folded paper towel and wipe out the oil like you would when seasoning it.