Heat your oven to 200°C/Fan 180°C/Gas 6. Finely dice the tomatoes. Finely chop the chilli. Flick out the seeds for less heat. Place the tomatoes and chilli in a small pan. Add 100ml water. Pop on a medium heat for 20 mins. Stir now and then.
Tip the flour into a large bowl. Scoop out 3 tbsp flour and put in a small bowl. Add 1 tbsp crème fraîche to the large bowl with a pinch of salt. Add 2 tbsp cold water and bring together to form a smooth dough. Add a little more water if needed. Dust with 1 tbsp of the reserved flour. Cover and set aside.
Peel and grate the carrot. Finely shred half the lettuce. Set aside. Zest the lime into the remaining crème fraîche and stir with a pinch of salt. Squeeze the lime juice into the tomatoes. Add 1 tbsp sugar. Stir on a low heat for 10 mins, stirring now and then. Take off the heat when thick and syrupy.
Add a large pinch of cayenne pepper and salt and pepper to the reserved flour. Crack the egg into a bowl and beat with a pinch of cayenne pepper and seasoning. Tip the polenta into a separate bowl. Dip a chicken fillet in the spiced flour and flip to evenly, but lightly, coat it. Dip it in the beaten egg. Roll it in the polenta. Put on a plate. Repeat with the remaining fillets.
Set a pan on a high heat for 1 min. Add 1 tbsp oil. Add half the chicken. Fry for 2 mins on each side, then place on a lined baking tray. Repeat with the remaining chicken, adding a little more oil if you need it. Slide into the oven. Cook for 5 mins to finish cooking the chicken. Check they are cooked by cutting into the thickest part of a fillet. There should be no trace of pink. If there is, return to the oven for a couple more mins.
Wipe the pan clean. Divide the dough into 2 balls. Roll thinly into 2 large circles. Heat the pan. Place 1 wrap into the pan and cook for 2 mins. Flip and sprinkle a little water on the cooked side. Cook for a further 2 mins. Repeat with the other wrap.
Put a wrap on each plate. Spread with crème fraîche. Top with carrot, lettuce and chicken. Add some chilli jam, then roll.