Gently flip the eggs and cook for about 30 seconds longer, or until the bottom sets but the eggs do not brown.
Sprinkle about a quarter of the bell pepper, a quarter of the basil, and a quarter of the cheese just off center on the omelet. Fold in half, cook for about 1 minute to soften the cheese and warm the bell pepper, and slide from the pan onto a plate and serve. Repeat to make 3 more omelets.
To roast bell peppers, char them over a grill or gas flame or under a broiler until blackened on all sides and soft. Turn them as they char to ensure even blackening. Remove from the heat and transfer to a bowl. Cover with plastic wrap and set aside for about 20 minutes to steam as they cool. Lift the peppers from the bowl and rub or peel off the blackened skin.