Season the chicken breasts with salt and pepper on both sides.
Heat olive oil and butter in a large skillet over medium-high heat. Once hot, add the chicken breasts and cook until golden brown on both sides, about 5-6 minutes per side. Make sure the chicken is cooked through (internal temperature should reach 165°F). Remove the chicken from the skillet and set aside.
In the same skillet, add minced garlic and sauté for about 1 minute until fragrant, being careful not to burn it.
Pour in the chicken broth and allow it to simmer for a couple of minutes, scraping up any browned bits from the bottom of the skillet.
Reduce the heat to low and stir in the heavy cream, Asiago cheese, and Italian seasoning. Whisk until the cheese has melted and the sauce is smooth.
Return the chicken to the skillet, spooning the sauce over the breasts. Simmer for an additional 5 minutes to allow the flavors to meld.
Garnish with chopped parsley and extra Asiago cheese before serving.