Season both sides of the chicken breasts generously with salt.
In a large skillet, heat olive oil over medium-high heat. Add the seasoned chicken breasts and sauté for about 6-7 minutes on each side until golden-brown and cooked through.
Transfer the chicken to a plate and cover to keep warm. In the same skillet, reduce heat to medium, add unsalted butter, and melt before pouring in the heavy cream.
Stir in the grated Pecorino Romano cheese and freshly cracked black pepper into the melted butter and cream mixture, whisking until smooth and creamy.
Return the sautéed chicken to the skillet, coating each piece in the creamy sauce. Simmer for an additional 2-3 minutes to meld flavors.
Taste the sauce and adjust with additional salt if needed before plating. Garnish with more black pepper and chopped parsley, and serve warm.