Preheat oven to 400 degrees. With a pastry brush, brush bottom and sides of a 9-by-13-inch baking dish with oil. In a large skillet, heat 1/4 cup water over medium-high. Working in batches, add spinach and cook, tossing, until completely wilted, about 8 minutes. Transfer to a colander to drain, pressing out as much water as possible. Return skillet to heat and add 1 tablespoon oil. Add leeks and cook until softened, about 8 minutes. Stir in garlic, season with salt and pepper, and cook until fragrant, about 1 minute. Stir in nutmeg. Transfer spinach and leek mixture to a food processor; pulse until coarsely chopped.
Cut phyllo sheets to fit baking dish. Place 1 sheet in dish (keeping remaining sheets covered with a damp towel) and brush lightly with oil. Stack 7 more phyllo sheets on top, brushing each with oil. Spread half the spinach mixture over phyllo and evenly sprinkle half the feta on top. Add 8 more phyllo sheets, brushing each with oil. Spread with remaining spinach mixture and feta. Top with remaining 8 phyllo sheets, brushing each with oil.
Bake until phyllo is golden and crisp on top, about 30 minutes. Let cool 15 minutes before cutting into squares. Serve warm or at room temperature.