- 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.