Eggplant and Walnut Phyllo Pie
['3 (1-pound) eggplants, peeled and cut into 1/2-inch-thick rounds', 'About 1 1/2 cups olive oil, divided', '3 medium leeks (white and pale green parts only)', '2 cups coarsely grated _gravié_ra or sharp white Cheddar (5 oz)', '3/4 cup coarsely grated Kefalotyri or Pecorino Romano (2 1/2 oz)', '1 1/2 cups walnuts (5 1/2 ounces), chopped', '1 1/2 teaspoons ground cumin', '20 (12- by 7-inch) phyllo sheets', 'thawed if frozen']
Sprinkle eggplant with 1 1/2 teaspoons salt and drain in a colander 30 minutes; pat dry.
Heat 3 tablespoons oil in a large nonstick skillet over medium-high heat until hot, then fry eggplant in batches, adding about 3 tablespoons oil per batch and turning once, until tender and golden brown, 5 to 7 minutes total per batch. Drain on paper towels and cool to room temperature. Wipe skillet clean.
Preheat oven to 375°F with rack in middle.
Halve leeks lengthwise and thinly slice. Wash and drain.
Cook leeks in 2 tablespoons oil in skillet over medium heat until golden, 7 to 9 minutes. Transfer to a bowl.
Chop eggplant; add to leeks with cheeses, nuts, cumin, and salt and pepper to taste.
Trim phyllo sheets into 9-inch squares. Cover stack of phyllo with plastic wrap and a damp kitchen towel. Layer 10 sheets in a 9-inch square baking dish, lightly brushing each sheet with oil. Spread filling on top, then cover with remaining phyllo, brushing each sheet with oil.
Cut several slits in phyllo (to vent) and bake until puffed and golden brown, 45 to 50 minutes. Let stand 10 minutes.
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