Slice each chicken breast lengthwise to make 4 thinner cutlets and pat them dry.
Season both sides of the cutlets evenly with 1/2 teaspoon kosher salt, 1/4 teaspoon black pepper, and 1 teaspoon Italian seasoning.
Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium heat.
Brown the chicken cutlets in the skillet about 4–5 minutes per side until golden and cooked through, then transfer to a plate.
Add the remaining 1 tablespoon olive oil to the skillet, then add the minced garlic and sauté until fragrant, about 30–60 seconds.
Pour in the tomato sauce and add the remaining 1 tablespoon butter. If using half-and-half, whisk the 2 teaspoons cornstarch into the 1/2 cup half-and-half until smooth, then stir into the skillet.
Return the chicken to the skillet, spoon sauce over the pieces, then sprinkle the 1/2 cup grated Parmesan over the top.
Simmer gently for about 5 minutes until the sauce thickens slightly and the chicken is heated through.
Serve the chicken topped with chopped parsley as a garnish.