Prepare a grill for medium heat. Toss zucchini, eggplant, and 1/4 cup oil on a rimmed baking sheet; season with 1 tsp. salt and 1/2 tsp. pepper.
Grill, turning often, until steamy, tender, and charred all over, 8-12 minutes. Return to baking sheet and let cool.
Cook pasta according to package directions.
Slice grilled vegetables into bite-size pieces and transfer to a large bowl. Add tomato, cheese, vinegar, thyme, and 1 1/2 tsp. salt, 1/2 tsp. pepper, and 1/2 cup oil and mix to combine. Drain pasta and immediately add to bowl with vegetables. Mix well to combine, then top with basil.
Do Ahead: Vegetables can be grilled 3 days ahead. Transfer (whole) to an airtight container and chill.
