Preheat oven to 350°F.
Heat the ghee over medium heat. Add chopped onions and sauté, stirring often, until translucent (about 20 minutes).
Slice the squash, zucchini and eggplant approximately ¼ inch thick (a mandoline works well). Place in a large bowl, drizzle with a portion of your olive oil (about ½ cup) and season with salt and pepper. Carefully toss to coat.
Slice the tomatoes using a serrated knife, drizzle with the remaining ¼ cup of olive oil and season with salt and pepper.
Spread onions evenly in a 12-inch round baking dish, cast-iron skillet or any oven-safe dish of a similar size.
Layer the eggplant on top of the onions, slightly overlapping each slice, until the entire dish is covered. Repeat this method with the zucchini and then the squash. Reserve any olive oil the remaining in the veggie bowl.
Grate the Parmigiano-Reggiano over the top of the squash in an even layer and sprinkle the pulled thyme over the cheese.
Layer the sliced tomatoes as you did the other vegetables. Place these carefully as this will be the top of the dish when it’s presented at the table.
Place on the middle rack of oven for 1 hour. The tomatoes should slightly brown but not burn or completely dry up. If they’re not slightly browned, bake for another 15–30 minutes. Your veggies will be done when your they are completely tender. To test, there should be no resistance when pierced with a knife.
Remove from oven. Add the final layer of cheese and drizzle with the remaining oil. Place back into the oven for roughly 5 minutes to melt the cheese.
Garnish just before serving with the fresh chiffonade of basil.