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. Place it in a bowl. Add 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 baby leaf salad into a bowl. Add 2 tsp olive oil and some salt and pepper. Toss to coat in the oil and seasoning. Put to one side for later.
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 chicken, ¼ of the pepper and ¼ of the cheddar on half the tortilla. Top with a few strands of sorrel and some of the red onion. Use a spatula to fold the tortilla over the filling so you have a half circle shell.
Cook the quesadilla for 1-2 mins till just golden underneath. Carefully turn over. Cook for another 1-2 mins till golden. Keep warm in the oven while you cook three more quesadillas. Serve them with the salad.