Preheat the oven to 375°F (190°C). Spray an 8×11½-inch baking pan with nonstick cooking spray.
If the chicken breasts are uneven in thickness, place each between sheets of plastic and pound to an even 3/4–1 inch thickness using a meat mallet.
Sprinkle both sides of the chicken breasts with 1 cup of the grated Parmesan and the black pepper, then arrange the breasts in the prepared baking pan.
In a small bowl, whisk together the Caesar dressing and the sour cream (if using) until smooth.
Pour or spoon the dressing mixture evenly over each chicken breast, then sprinkle the remaining 1/2 cup grated Parmesan over the top.
Bake at 375°F for about 30 minutes, or until an instant-read thermometer inserted into the thickest part reads 150°F.
Switch the oven to broil and broil the chicken 2–4 minutes until golden brown, watching closely to avoid burning; final internal temperature should reach 165°F.
Remove from the oven, tent loosely with foil, and let the chicken rest 5–10 minutes before serving. Garnish with chopped parsley if desired.