Preheat the oven to 425 F.
Dry chicken with paper towel.
Season on both sides with salt and pepper.
Melt the butter over medim high heat in a large cast iron skillet.
When butter is hot, add chicken breasts and sear 4 minutes per side.
While chicken is cooking, cut potatoes in half (or quarters, if large) and toss in a large bowl with olive oil, salt, and pepper.
Place potatoes around chicken cut side down and transfer the skillet in the preheated oven.
Cook for 20 minutes or until chicken has reached 170 F and potatoes are soft when pierced with the knife.
Transfer the chicken and potatoes to a platter.
Preheat the skillet for a minute over medium heat (it will still be very hot).
Add heavy cream and scape off any stuck pieces.
Add Parmigiano Reggiano, stir, and taste for salt. Adjust the seasoning as needed.
Add lemon slices, stir, and cook over medium heat for about 3 minutes as the sauce thickens. Taste for seasoning.
Pour the sauce over chicken and potatoes on a platter or place the chicken and potatoes back in the skillet and mix with the sauce.
Top with freshly ground black pepper and chopped parsley, if desired.
Serve and enjoy!