Chicken with Red Pepper Sauce

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Chicken:

6 chicken breast fillets, flattened to 1/4-inch-thick

3 tablespoons olive oil 

1 tablespoon kosher salt 

2 teaspoons freshly ground black pepper 

1 tablespoon Italian seasoning 

Red Pepper Sauce:

2 tablespoons salted butter

1 tablespoon olive oil

1/2 large onion, finely diced 

3 cloves garlic, minced 

1 jar (15.5 ounces) roasted red peppers, drained and roughly chopped 

1 to 2 cups vegetable or chicken broth 

1/2 teaspoon kosher salt, more to taste 

Freshly ground black pepper 

1/2 cup heavy cream (or more to taste) 

One 4-ounce chunk Parmesan, to shave 

2 tablespoons finely minced parsley