Cut chicken breasts into 3/4-inch (2cm) strips. Cut the bell peppers and onion into thin slices (about 1/4-inch (5mm)). Add into a large bowl.
Mix Fajita seasoning with 2 tablespoons of oil, toss with chicken and vegetables until coated on all sides. Season with salt and pepper.
Heat the remaining oil in a large frying pan (preferably a non-stick pan). Cook the chicken with vegetables in 3 batches over high heat until the chicken is nicely browned on both sides and vegetables slightly soft but still crunchy. Transfer into a very large bowl, leave until slightly cooled (you can shred the cheese on the big holes of a box grater while they are cooling).
Add the shredded cheese and lime juice to the bowl with chicken and vegetables, stir everything together.
Place about 3 tablespoons of filling on one half of the tortilla, then fold the other half over to cover the filling. Don't overfill the tortillas or they will be difficult to eat (the filling will fall out).
Heat a small amount of oil in a large frying pan over medium heat. When hot, add two folded in half tortillas. Cook on both sides until tortilla is browned on the outside and the cheese is melted (about a minute per side).
Transfer to a chopping board, cut in half.
Serve with tomato salsa or yogurt lemon dipping sauce (recipes are below in the notes section) or just on their own.