Pour the buttermilk into a wide, shallow bowl. Cut each chicken breast in half horizontally and pat dry with kitchen paper. Turn them over in the buttermilk, leave in the dish and chill for 1 hr. Meanwhile, for the salad, mix the carrot with the lemon juice and olive oil. Mix the sauce ingredients together in another bowl and set both aside.
Heat a large, non-stick frying pan, tip in the breadcrumbs and toast in the dry pan for 2-3 mins until pale brown, stirring regularly so they don’t burn. Tip into a bowl and break up any large clumps.
Heat oven to 230C/210C fan/gas 8. Tip the crumbs onto a large plate, then lift the chicken breasts from the buttermilk, and roll them in the crumbs, making sure they are coated evenly.
Heat ½ tbsp of oil in a non-stick frying pan. Fry the chicken for 1-2 mins over a medium-high heat, then turn it over, adding the rest of the oil and fry for 1 min more.
Transfer the chicken to a baking sheet and bake for 5 mins, then rest for 5 mins and cut into strips. Serve the chicken and the carrot salad with the sauce and baked potatoes.