Phyllo dries out very fast and becomes brittle when exposed to the air. Check out our test kitchen tips for working with it.
Position a rack in the center of the oven and heat the oven to 375°F.
Make the filling: Heat a 10-inch straight-sided saute pan over medium-high heat. Add a few large handfuls of the spinach and cook, tossing gently with tongs. As the spinach starts to wilt, add the rest a few handfuls at a time. Cook until all the spinach is wilted and bright green, about 4 minutes. With a slotted spoon, transfer the spinach to a colander set in a sink. Let cool slightly and squeeze with your hands to extract as much of the remaining liquid as you can.
Wipe the pan dry with a paper towel. Heat the oil in the pan over medium heat. Add the scallions and cook until soft and fragrant, about 4 minutes. Stir in the spinach, turn off the heat, and let cool for 5 minutes. Then stir in the cheeses, eggs, dill, parsley, nutmeg, and 1/2 tsp. salt and mix thoroughly.
Assemble the pie: With a pastry brush, lightly coat the bottom and sides of a 9x13x2-inch baking pan with some of the oil. Working quickly, lightly oil one side of a phyllo sheet and lay it in the pan oiled side up and off center so that it partially covers the bottom and reaches halfway up one long side of the pan (the edge on the bottom of the pan will be about 1 inch from the side). Lightly oil the top of another phyllo sheet and lay it oiled side up and off center so it reaches halfway up the other long side of the pan. (If your pan has sloped sides, the sheets may be slightly longer than the bottom of the pan; if so, let the excess go up one short side of the pan and then alternate with subsequent sheets.) Repeat this pattern with 4 more phyllo sheets.
Next, lightly oil the tops of 3 phyllo sheets and layer them oiled side up and centered in the pan. Spread the filling evenly over the last layer.
Repeat the oiling and layering of the remaining 9 phyllo sheets over the filling in the same way you layered the previous 9. With the oiled bristles of the pastry brush, push the edges of the phyllo down around the sides of the pan to enclose the filling completely.
With a sharp knife, score the top phyllo layer into 24 rectangles, being careful not to cut all the way through to the filling. Using the same pastry brush, brush the milk along all the score marks (this will keep the phyllo from flaking up along the edges of the squares). Bake the spanakopita until the top crust is golden brown, 35 to 45 minutes. Let cool until just warm. Cut out the rectangles carefully along the score marks and serve.
You can make the pie up to 4 hours ahead. Keep warm, if desired, or serve at room temperature.