Cook the fettuccine according to package directions, reserve ¼ cup of pasta cooking water, drain, and set aside.
Season the chicken breasts on both sides with 1 ½ teaspoons cajun seasoning, salt and pepper. Heat the butter and half of the olive oil in a large skillet and brown the chicken breasts for about 4-5 minutes on each side until cooked through. Transfer to a plate and keep warm.
Add the remaining olive oil to the same skillet and saute the chopped onion and minced garlic for about 2 minutes, scraping the bottom of the pan to release the bits stuck on the bottom.
Pour in the heavy cream, sun-dried tomatoes, basil, parsley, sliced green onions, 1 1/2 teaspoons Cajun seasoning, salt, and pepper to taste. Simmer the sauce until slightly thickened.
Slice the cooked chicken breasts and return them to the skillet along with the cooked fettuccine. Stir everything until heated through. Add a little bit of pasta cooking water if the sauce is too thick. Serve with freshly grated Parmesan on top.