Heat oven to 375°F.

Prepare your pan: Line a deep 10- to 11-inch ovenproof skillet or cast-iron pan or cake pan with a large round of parchment paper, pressing it across the bottom and pleating it up the sides.

Prepare filling: Set aside ¼ cup sliced green onions. Toss remaining green onions, all the parmesan, flour, thyme, 1 ½ teaspoons kosher salt and lots of freshly ground pepper (about ¾ teaspoon) in a medium bowl to blend.

Assemble torte: Layer ⅓ of the potatoes in concentric circles in the bottom of your prepared pan, overlapping them slightly. Layer ½ of squash in concentric circles atop the potatoes. Drizzle with 2 teaspoons of oil and sprinkle with ⅓ of the cheese mixture. Repeat with ⅓ potatoes, ½ of squash, 2 teaspoons oil, and ⅓ cheese mixture. Top with the last ⅓ of potatoes, drizzle with remaining 2 teaspoons of oil, and sprinkle with remaining cheese mixture. Press gently to flatten.

Bake torte: Cover pan tightly with foil (or with a lid, if your pan has one) and bake for 45 to 50 minutes, or until potatoes are almost tender. Remove foil (or lid) and bake another 30 to 35 minutes, until the torte is browned on top and the potatoes are tender.

To serve: Let cool for 10 minutes in pan then use the sling of parchment paper to lift the torte out of the pan and onto a serving plate or cutting board. Sprinkle with reserved green onions. Cut into 8 wedges and eat right away.

Do ahead: The torte can be made in advance and kept chilled in the fridge until needed. To rewarm, cover again with foil or lid, and place in a 350-degree oven until heated through, about 30 minutes.