Heat grill to high. Put the chicken onto a roasting tray, coat with the oil, season, then grill for about 12 mins, turning halfway through the cooking time.
Meanwhile, boil the pasta, adding the beans for the final 3 mins, then drain, reserving 150ml of the cooking water.
Shred the cooked chicken and put into the pasta pan with the crème fraîche, lemon juice, 4 tbsp Parmesan and the reserved cooking water. Heat gently, season to taste, then stir in the parsley and pasta. Serve sprinkled with the remaining Parmesan.