Prepare a medium fire on a gas grill or a medium-hot charcoal fire. In a large bowl, toss the tomatoes and zucchini with 2 tablespoons of the oil, 2 teaspoons of the thyme, and the salt and pepper.

Set the vegetables cut side down on the grill and cook without moving them until they have good grill marks, 5 to 7 minutes. Flip and cook until browned and tender, 6 to 8 more minutes. Transfer to a cutting board and let cool for a couple of minutes. Coarsely chop, return them to the same large bowl along with 1-1/2 cups of the pecorino, and toss. Let sit for up to a couple of hours at room temperature.

Bring a large pot of well-salted water to a boil over high heat. Add the pasta and cook, stirring often, until just al dente, about 11 minutes. Drain well and toss with the tomato mixture, 3 tablespoons of the chives, the remaining 3 tablespoons olive oil and the balsamic vinegar. Season generously with salt and pepper to taste and transfer to a serving bowl. Sprinkle with the remaining 1 tablespoon chives, 2 teaspoons thyme, and 1/2 cup pecorino, and serve.

Make Ahead: You can grill the tomatoes and zucchini a couple of hours before serving. Hold them in a large bowl at room temperature. Cook the pasta just before guests arrive and toss it with the vegetables.