How to Make It
Set the chicken skin side up on a rack set over a baking sheet. Press down firmly on the breastbone to flatten the bird. Season the chicken all over with 2 teaspoons of salt and refrigerate uncovered overnight. Let the chicken stand at room temperature for 30 minutes before roasting.
Preheat the oven to 425°. Rub the chicken with 2 tablespoons of the olive oil and season with pepper. Roast for about 1 hour, until golden and an instant-read thermometer inserted in the thigh registers 160°. Transfer to a rack and let rest for 15 minutes.
Meanwhile, in a large bowl, whisk the vinegar with the honey and mustard. Whisking constantly, drizzle in the remaining 1/2 cup of olive oil and season with salt and pepper. Transfer half of the dressing to a small bowl. Add the peaches to the large bowl and toss to evenly coat. Let the peaches marinate at room temperature for 30 minutes.
Cut the chicken into 8 pieces and arrange on a platter. Add the arugula and basil to the marinated peaches and toss to coat. Transfer the peach salad to a bowl, top with the chopped peanuts and serve with the chicken, passing the reserved dressing at the table.