Season the chicken breasts on both sides with salt and pepper.
Heat the olive oil and butter in a large skillet over medium-high heat.
Sear the chicken breasts for 5 to 7 minutes per side until golden brown and cooked through (internal temperature reaches 165°F or 74°C). Remove the chicken from the skillet and set aside.
Reduce the heat to medium. Add the minced shallot to the same skillet and cook for 1 minute until softened.
Add the minced garlic and cook for 30 seconds until fragrant. Do not let the garlic burn.
Pour in the chicken broth and scrape up any browned bits from the bottom of the skillet. Let the broth simmer and reduce slightly for 2 minutes.
Stir in the heavy cream and Italian seasoning. Bring the mixture to a gentle simmer.
Remove the skillet from the heat. Gradually stir in the grated Asiago cheese and Parmesan cheese until the sauce is smooth and melted. Do not boil the sauce after adding the cheese, as this can cause it to separate.
Return the cooked chicken breasts to the skillet. Spoon the sauce over the chicken.
Garnish with fresh parsley before serving.
Notes
For a thicker sauce, let the broth reduce longer before adding the cream.
If the sauce seems too thick, add a splash more chicken broth or cream to reach your desired consistency.
Serve this dish over pasta, rice, or alongside steamed green beans.