Preheat the oven to 425 degrees. Grease a 9×13-inch ceramic or glass baking dish with 1 tablespoon of the olive oil, and rub the sides with the garlic clove halves. Set aside.
Using a mandoline, thinly slice the zucchini and pile into the baking dish. Drizzle with the remaining 2 tablespoons olive oil and season well with salt and pepper; toss to coat. Gently press the zucchini into the dish and sprinkle with the thyme sprigs. Cover with aluminum foil and bake for 30 minutes.
Remove the foil and discard the thyme springs. Crumble half the goat cheese over the zucchini and season again with salt and pepper, stirring to combine. Gently press down again with a spatula. Sprinkle with half of the basil, the remaining goat cheese, and the Parmesan. Bake, uncovered for 30 more minutes, or until the liquid has evaporated, and the top is golden brown. Let cool for 20-30 minutes and garnish with the remaining fresh basil. Serve warm or at room temperature.
MAKE AHEAD: Make this a day ahead and cool completely. Store, covered, in the refrigerator, and reheat at 375 until warm.