Grilled Chicken with Peach Salad

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Vegetable oil, for the grill

Four 6- to 8-ounce boneless, skinless chicken breasts

Kosher salt and freshly ground black pepper

Zest and juice of 1/2 lemon

2 ripe peaches, pitted and cut into wedges

2 large tomatoes, preferably heirloom, cored and cut into wedges

1/4 cup lightly packed basil leaves, torn

2 tablespoons roughly chopped chives

1 tablespoon extra-virgin olive oil, plus extra for drizzling