Season the halved chicken breasts evenly with the salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat. Sear the meat for 5-6 minutes per side until golden brown and a meat thermometer reads 165°F. Remove from the skillet and set aside on a plate.
Reduce the heat to medium. Add the butter to the same skillet. Once melted, add the sliced mushrooms and saute for 5 minutes until browned and tender.
Stir in the minced garlic and cook for 1 minute until fragrant.
Pour in the chicken broth to deglaze the pan, scraping up any browned bits from the bottom. Allow it to simmer and reduce for 1-2 minutes.
Reduce the heat to low. Whisk in the heavy cream and grated Parmesan cheese. Simmer gently for 3-4 minutes until the sauce slightly thickens.
Return the seared meat (and any resting juices) to the skillet. Spoon the creamy sauce over the top and simmer for 1 more minute to heat through.
Garnish with chopped fresh parsley and serve immediately.