Preparation

• Place a rack in the center of the oven, and heat the oven to 350°F. Rub a 4-quart shallow gratin or other shallow baking dish (a 15x10x2-inch Pyrex dish is fine) with a little butter.
• In a medium bowl, combine the breadcrumbs, 1/4 cup of the Parmigiano, 2 Tbs. of the oil, 2 tsp. of the thyme, and 1/2 tsp. salt. Stir well to combine, and set aside.
• In a large, heavy-duty nonstick skillet, melt the 2 Tbs. butter with 2 Tbs. of the oil over medium-low heat. Add the leeks (it’s okay if some water clings to them) and 1/2 tsp. salt. Cover and cook, stirring occasionally, until the leeks are softened and translucent, about 5 minutes. Uncover, raise the heat to medium, and continue to cook, stirring frequently, until the leeks are shrunken and browned in places, about 6 minutes more. Add 2 tsp. thyme and the garlic, and stir until fragrant, about 30 seconds. Carefully add all of the spinach to the pan, and toss with the leeks until the spinach is wilted, about 1 minute. Transfer the leeks and spinach to the gratin dish, and use a spatula to smooth into an even layer.
• Put the sliced squash, the remaining 4 Tbs. olive oil, the remaining 3-1/2 tsp. thyme, and 2 tsp. salt in a mixing bowl. Add several grinds of pepper, and stir until the squash is thoroughly coated.
• Beginning at one end of the gratin dish, arrange a row of slightly overlapping squash slices, slightly propped up against the edge of the pan. Sprinkle with Parmigiano, using about 1-1/2 Tbs. per row. Shingle another row of squash slightly overlapping the first, sprinkle with Parmigiano, and continue arranging until the pan is full. If you think you’re going to come up short, push down on the rows with the palm of your hand to spread them out somewhat. (If you think you are going to have a lot of extra squash, you can push the rows back to make room for more. You’ll usually wind up with a few extra pieces.)
• Drizzle whatever liquid is left in the mixing bowl over the vegetables, and sprinkle them with any remaining Parmigiano. Drizzle with a bit of olive oil. Cover the top with the breadcrumb mixture. Bake until the squash is tender and the top of the gratin is deep golden ( juices will bubble and darken around the edges), about 45 minutes, rotating the pan halfway through baking for even browning. Let sit at least 15 minutes before serving.