Use a sharp knife to slice both chicken breasts in half through the middle, resulting in 4 thinner chicken cutlets. Season with salt and pepper.
Heat a heavy skillet over medium to medium-high heat and add the butter and olive oil. Cook the chicken until it has nice color on the outside and is cooked through, about 2 1/2 minutes per side. Remove the chicken to a plate.
Pour the marinara sauce into the skillet and cook, stirring, to heat through. Nestle the chicken cutlets in the sauce, then arrange some pepperoni slices over the top of each one. Lay some cubed mozzarella on top, then cover the skillet with a lid and cook until the cheese is melted, 2 to 3 minutes. Sprinkle with the parsley.
Serve immediately with a side salad or over pasta, with marinara spooned over the top.