Heat the oven to 375°F. Cut the spaghetti squash in half length-wise and scoop out the seeds. Brush the flesh with 2 tablespoons of olive oil and sprinkle with a little salt and pepper. Place the squash, cut side down, on a rimmed baking sheet and bake until tender, about 45 minutes. Remove the squash from the oven and let it stand until it is cool enough to handle. Scrape the flesh into a bowl and set aside. Leave the oven on.

While the squash is baking, bring the vegetable broth to a boil in a saucepan placed over medium-high heat. Carefully pour in the rice. When the liquid returns to a boil, reduce the heat to medium-low and cover. Cook the rice for about 15 minutes, until the grains are slightly underdone and the liquid has been absorbed. Transfer the rice to a large bowl and toss with 1 tablespoon of olive oil.

Put the remaining 3 tablespoons of oil and the garlic in a large skillet or sauté pan and set the pan over medium-low heat. Sauté the garlic until it is softened but not at all browned, about 5 minutes. Add the mushrooms, onion, and zucchini and sprinkle with a little salt and pepper. Sauté the vegetables for about 10 minutes, until they are tender and the liquid they release has been reabsorbed. Stir in the parsley. Remove from the heat and transfer to a bowl. Let cool for a few minutes.

Add the reserved spaghetti squash and the mushroom-zucchini mixture to the bowl with the rice. Add the dry Jack and mozzarella cheeses and gently but thoroughly fold everything together.

Grease an 8-by-12-inch or similar-sized gratin dish with a little olive oil. Pour the rice-vegetable mixture into the dish and sprinkle the Parmigiano on top. Bake the gratin for 20 to 25 minutes, or until the top is golden brown and the edges are crusty. Remove the gratin from the oven and let it rest for 5 to 10 minutes before serving.