Juice the lime. Measure out 2 tsp caster sugar and stir it into the lime juice with a pinch of salt. Peel and very finely slice the red onion. Add it to the lime juice. Stir to mix well. Put to one side to pickle.
Slice the chicken breast fillets into long, thin strips that are all roughly the same size. Pop them in a separate bowl. Add 2 tsp of the Cajun seasoning. Toss to mix. Put to one side to marinate for 10 mins.
Halve the pepper. Scoop out the seeds and white bits. Slice the pepper into long, thin fingers. Coarsely grate the chunk of cheddar. Finely slice the sorrel leaves.
Put the salad into a bowl. Add 2 tsp oil and some salt and pepper. Toss to coat in the oil and seasoning. Set aside.
Warm 2 tsp olive oil in a frying pan. Add the Cajun chicken. Fry for 5-8 mins, turning now and then, till the chicken is golden brown and cooked through. Lift out of the pan. Put on a plate.
Add the pepper to the pan. Cook and stir for 2-3 mins till the pepper is soft and a little charred. Add to the chicken. Wipe the pan clean with kitchen paper. Set your oven to its lowest setting.
Put the pan on a medium heat. Add a tortilla to the pan. Heap ¼ of the cheddar, ¼ of the pepper and ¼ of the chicken on half the tortilla. Top with a few strands of sorrel and some red onion. Cook the quesadilla for 2-3 mins till the cheese melts.
Use a spatula to fold the tortilla over the filling so you have a half circle shell. Carefully slide the quesadilla onto a plate or serving dish. Keep warm in the oven while you cook 3 more quesadillas. If you have any leftover pepper or red onion, add them to the salad. Serve the quesadillas with the salad.