Prepare the ingredients: Cut the chicken into 1-inch (2.5cm) wide strips. Cut bell peppers into strips and onion in half then into slices.
In a medium bowl, stir together Fajita seasoning and 1 tablespoon of oil. Add the chicken, season with salt and pepper, and toss to coat.
Make the sauce: add all the ingredients to a food processor and process until smooth. Season with salt and pepper to taste.
Cook the chicken and vegetables: In a large frying pan, heat the remaining 1 tablespoon of oil over medium-high heat. Add the chicken strips in an even layer and cook until browned. Turn over and cook until cooked through. Transfer to a plate and drizzle immediately with lime juice.
Add the bell peppers and onion in an even layer to the pan. Cook for 2-3 minutes until lightly browned and softened but still crunchy. Season with salt and pepper and transfer to a plate.
Assemble the wraps: Place 4 slices of cheese on each wrap, top with the chicken and vegetables. Drizzle over the juices from the chicken mixed with lime juice (from the plate where the chicken was). Spoon about 1-2 tablespoons of avocado sauce on each wrap.
Wrap like a burrito. Place the wrap in a sandwich/panini press until warm. You could also reheat it in the oven or in a pan, just to melt the cheese.
Enjoy!