Add the chicken breasts to a tupperware dish, and pour the dressing on top of them. Add the lid, closing to seal, and refrigerate them-turning the dish over every 30 minutes for 2-4 hours.
Add half of the olive oil to a large cast iron skillet set over medium heat. Once the oil's hot (shimmery) add the onion & peppers to the skillet, tossing gently to coat in the hot oil.
Cook, stirring often, until the peppers have just begun to soften & the onions caramelize. Season with salt & pepper, to taste. Using a slotted spoon, transfer the onion & peppers to a waiting plate and cover the plate tightly with foil. Set aside.
Add the rest of the oil to the hot skillet, still set over medium heat. Let it heat up for several seconds.
Using a pair of tongs, remove the chicken from the marinade (let the excess drip off) and add the chicken breasts to the hot skillet. Sear the chicken until nicely golden brown, then flip and sear the other side of each breast- roughly 2 minutes per side.
Turn the heat off and using thick pot holders, carefully transfer the skillet to the oven. Bake at 400°for 15 minutes, or until a meat thermometer inserted in the thickest part of each breast reads 165°F.
Carefully remove the hot skillet from the oven, and set it aside. Turn off the oven, and set a flat cast iron skillet inside of the still hot oven allowing it to preheat for 5-8 minutes.
Carefully remove the hot skillet and set it on a wooden liner or thick trivet.
On one side of the skillet, spread half of the American cheese slices down. Sprinkle the shredded Monterrey cheese out overtop of the slices, then top with the remaining American cheese.
Spoon the pepper and onion mixture evenly out over the cheese.
Add mashed potatoes to the other half of the flat skillet, season them as desired. Top the pepper and onion mixture with the cooked chicken & serve the skillet meal immediately.