Heat the olive oil in a skillet with a tight-fitting lid over medium heat. While the oil heats, season the chicken with salt and pepper. Add the chicken to the skillet and cook for 5-7 minutes on one side until it is browned well. Flip the chicken, add the water, and cover with the lid. Let it cook for 8-10 minutes until fully cooked. Set the chicken aside to rest and discard the liquid in the pan. Once cooled chop or shred the chicken.
In a large bowl, combine the pesto, mayo, salt and pepper to taste, and the roasted red peppers. Stir until combined, then add the chicken and stir until fully coated.
Store ingredients separately OR assemble the sandwiches! To assemble, place a slice of cheese on the bottom of the sandwich, top with 1/3 of the pesto chicken salad, 1/3 cup of arugula, and the top slice of bread. Wrap it up and repeat two more times! Store the whole sandwich in parchment paper, beeswax wrap, plastic wrap, or aluminum foil. These hold up best if you assemble them right before eating. These can be eaten cold, room temp, or reheated in a microwave or oven. That's it!