Preheat the grill to medium and line the rack with foil. Measure 1 tbsp of the vinegar into a shallow dish with 1 tbsp olive oil and a good shake of salt and pepper. Add the chicken fillets and turn them in the dressing until they are well coated. Put the chicken in one layer on the foil-lined rack and grill for 10-12 minutes, then turn them and grill for a further 10-12 minutes until the chicken is cooked through, glossy and deep brown in colour.
Halve the peaches, remove the stones and slice fairly thickly. Spread the spinach leaves over a serving platter. Slice the cooked chicken and scatter it over the spinach leaves along with the peaches and half the basil leaves. Shred the remaining basil and mix in a small bowl with the remaining balsamic vinegar, 2 tbsp olive oil and some salt and pepper. Drizzle over the salad and serve.