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Preheat oven to 375°F. Butter two 8-inch-diameter cake pans. (Deb note: I had only a 9-inch pan around, so what you see in my pictures is slightly thinner.) Set aside 1/4 cup sliced green onions. Toss remaining green onions, cheese, flour, thyme, salt and pepper in medium bowl to blend.
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Layer 1/6 of potatoes in concentric circles in bottom of 1 prepared pan, overlapping slightly. Layer 1/4 of squash in concentric circles atop potatoes. Drizzle with 1 teaspoon oil. Sprinkle with 1/6 of cheese mixture. Repeat with 1/6 of potatoes, then 1/4 of squash and 1 teaspoon oil. Sprinkle with 1/6 of cheese mixture. Top with 1/6 of potatoes. Drizzle with 1 teaspoon oil. Sprinkle with 1/6 of cheese mixture and press gently to flatten. Repeat procedure with second cake pan and remaining potatoes, squash, oil, and cheese mixture.
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Cover pans with foil. Bake until potatoes are almost tender, about 40 minutes. Remove foil; bake uncovered until tortes begin to brown and potatoes are tender, about 25 minutes longer. (Can be made 6 hours ahead. Cool. Cover with foil and chill. Rewarm, covered with foil, in 350°F oven until heated through, about 30 minutes.)
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Cut each torte into wedges. Sprinkle wedges with 1/4 cup green onions; serve.