Put the olive oil, Parmesan, pine nuts, oregano, garlic, basil, mint, parsley and some salt and pepper in a blender and puree. Put the chicken breasts in a resealable plastic bag, pour in the marinade and seal. Refrigerate for 30 minutes to several hours, the longer the better.
Preheat a grill pan over medium-high heat and brush with oil. Remove the breasts from the marinade and grill until cooked through with nice char marks, about 4 minutes a side. Slice the chicken and serve over Fettuccine Alfredo. Garnish with more basil.
Bring a pot of water to a boil and cook the pasta according to the package directions.
Meanwhile, in a saucepan or skillet over low heat, add the cream and butter and cook until the butter melts and the mixture is warm. Season with salt and pepper.
Put half of the Parmesan into a large serving bowl. Pour the warm butter/cream mixture over the top.
Drain the pasta (reserving a little of the cooking water) and immediately pour it into the bowl. Toss a couple of times, then sprinkle in the other half of the Parmesan. Toss to combine, thinning with the reserved pasta water if necessary.