Make the pesto: In a small food processor or blender process the basil, garlic, pine nuts, and ½ teaspoon salt. Then, with the motor running, drizzle ¼ cup oil through the feed tube until the mixture forms a thin paste, scraping the sides of bowl if necessary. Add the cheese and process briefly, just until combined. Set aside until ready to use; or transfer to a jar with a tight-fitting lid and refrigerate for up to 3 days. Bring to room temperature before serving.
Prepare the grill for direct cooking over medium heat (350º to 450ºF).
Cut the vegetables lengthwise into ¼-inch-thick slices. Drizzle with oil and season lightly with salt and pepper.
Brush the cooking grates clean. Grill the vegetable slices over direct medium heat, with the lid closed, until well marked and tender, 5 to 7 minutes, turning once. Remove from the grill.
Brush the cooking grates clean. Grill the vegetable slices over medium heat, with the lid closed, until well marked and tender, 5 to 7 minutes, turning once. Remove from the grill.
Arrange the vegetable slices on a warmed platter, overlapping them, and top with some of the pesto. Serve immediately with the remaining pesto.
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