Preheat a clean barbecue to high heat (425–450°F).
To grill: Brush the zucchini with the oil. Grill for about 3 minutes per side or until dark grill marks appear and the zucchini is still tender-crisp. Do not overcook, or it will get soggy as it cools. Transfer the zucchini to a plate to cool.
Cut the cooled zucchini on the bias into 1-inch pieces. If not preparing the salad immediately, transfer the zucchini to an airtight container and refrigerate for up to 2 days.
To serve: In a large bowl, combine the zucchini, tomatoes, and pesto. Season with salt and pepper and stir to combine. Arrange the salad on a serving plate.
Rip the cheese into bite-sized pieces and scatter on top of the vegetables. Garnish with the radishes and basil. Serve immediately.