Slice each chicken breast into 5-6 strips. Warm 1 tbsp oil in a large pan over a medium heat. Fry the chicken for 4-5 mins till golden. Transfer to a plate.
Peel and thinly slice the onion. Add to the pan and sweat for 3-4 mins.
Finely slice the red peppers and garlic. Halve the cherry tomatoes.
Tip the peppers, garlic and cherry tomatoes into the pan and season. Pour in 3 tbsp water. Cook on a low heat for 20 mins, till all the veg are tender.
Finely grate the cheddar and set aside.
Pour 750 ml boiling water into a separate pan and bring to the boil. Add a generous pinch of salt. Pour in the polenta in a constant trickle, stirring vigorously with a wooden spoon. Cook for 4-5 mins, stirring constantly, till it thickens and starts to pull away from the sides of the pan. Stir in the cheddar.
Return the chicken to the peperonata and warm through. Serve alongside dollops of the polenta. Scatter over some basil leaves and serve.