Preheat oven to 375°F. Butter two 8-inch-diameter cake pans. Set aside 1/4 cup sliced green onions. Toss remaining green onions, cheese, flour, thyme, salt and pepper in medium bowl to blend.
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.
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.)
Cut each torte into wedges. Sprinkle wedges with 1/4 cup green onions; serve.