1. Preheat oven to 350°F. Warm oil in a skillet over medium-high heat. Cook onion, stirring often, until softened, 3 to 5 minutes. Transfer to a large bowl.
2. Squeeze out as much moisture from spinach as possible. Add to bowl with onion. Stir in cheeses, bread crumbs, salt, pepper, nutmeg and eggs; mix well. (Can be made up to 2 days ahead to this point. Cover and chill.)
3. Butter a 9-by-13-inch baking pan. Unroll phyllo onto a clean counter; cover with lightly moistened paper towels. Line pan with 1 phyllo sheet, keeping remaining dough covered. Brush lightly with butter. Top with another sheet; butter again. Repeat with butter and phyllo for a total of 6 sheets.
4. Spread spinach mixture in pan. Fold edges of phyllo onto filling. Trim edges of remaining phyllo sheets to fit inside pan. Place 1 sheet of phyllo on top of filling and brush lightly with butter. Repeat with remaining phyllo and butter. Bake until browned and bubbly, about 45 minutes. Let cool for 15 minutes before serving.
Make It Ahead: To Freeze: Bake pie as directed, let cool completely, wrap in foil and freeze. To Serve: Bake, covered with foil, at 375ºF for 20 minutes. Uncover and continue to bake until heated through, 10 to 20 minutes longer. (If the pastry starts to brown too much, re-cover it with foil.)