Crack 3 eggs into a bowl. Separate the remaining 3 eggs and add the yolks to the whole eggs. Stir in the Parmesan and some of the basil and season to taste. Whisk the egg whites until soft peaks form, then carefully fold into the whole egg mixture, one-third at a time.
Heat the oil in an ovenproof frying pan. Add the egg mixture and cook for 2 minutes, then scatter over the peppers and goats' cheese.
Place the pan under a preheated hot grill and cook for 5–7 minutes more until puffed and just set. Scatter over the remaining basil to serve.