Pat chicken breasts dry with paper towels. Season both sides with Italian seasoning, Kosher salt and freshly cracked pepper.
Heat a large cast iron or stainless-steel skillet over medium-high heat. Add the olive oil and butter.
Once the butter is foaming, add the chicken. Cook for 3-4 minutes per side, until deeply golden and the thickest part of the chicken’s internal temperature reaches 165°F. Depending on the thickness of the chicken, you may need to reduce the heat to medium-low or low and cover the skillet with a lid to finish cooking.
Transfer the chicken to a plate and tent loosely with foil to keep warm.
Add the shallot to the skillet and cook for 2-3 minutes, until softened. Add the garlic and cook for 30 seconds until fragrant.
Stir in the chicken broth, scraping up the browned bits from the skillet. Simmer for 3-4 minutes to reduce the broth slightly.
Reduce heat to medium-low. Stir in the Boursin cheese until smooth and fully incorporated, then stir in the heavy cream. For a thicker sauce, make a cornstarch slurry by mixing together 1/2 tsp cornstarch and 1 tsp water in a small bowl. Stir the slurry into the sauce and simmer for 1-2 minutes, until slightly thickened.
Return the chicken to the skillet and spoon the sauce over each piece. Gently simmer for 2-3 minutes to warm the chicken and coat with the sauce.
Garnish with fresh chopped parsley and serve over mashed potatoes.