Season chicken cutlets on both sides with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat until shimmering.
Place chicken in the skillet and sear for 5 to 7 minutes per side until an internal temperature of 165°F (74°C) is reached. Remove chicken from the pan and set aside.
In the same skillet, add sliced mushrooms and sauté for 5 minutes until browned and the released liquid has evaporated.
Add minced garlic to the mushrooms and sauté for 60 seconds until fragrant.
Deglaze the skillet with chicken broth, scraping the bottom to release fond, then stir in the heavy cream and dried oregano.
Bring the liquid to a simmer over medium heat and stir in the Parmesan cheese.
Reduce heat to medium-low and simmer for 3 to 5 minutes until the sauce achieves a nappe consistency.
Return the chicken cutlets to the skillet, spooning the sauce over them to coat thoroughly.
Garnish with chopped parsley and serve immediately.