Season chicken cutlets with Italian seasoning, salt, and pepper. Dredge in flour, and shake off any excess flour.
In a skillet, melt butter and heat olive oil. When shimmering, add the chicken and cook the cutlets on both sides until cooked through. The chicken is done when the internal temperature reaches 165°F/74°C using a kitchen thermometer. This takes 4 to 5 minutes per side, depending on the thickness of the cutlets. Remove onto a plate and keep warm.
Add garlic, and cook for 30 seconds. Then add the tomato paste and the heavy cream and whisk until well combined. If the sauce is too thick, add a splash of water.
Add parmesan, and stir it in until melted, then add the tomato and let the sauce cook for 2 to 3 minutes. Have a taste and adjust salt and pepper to your preference.
Add the chicken back in, and spoon the sauce over the chicken. Remove from heat. Garnish with chopped basil and extra parmesan if desired.