1. Set a large saucepan over moderately high heat. Add large handfuls of the spinach, tossing constantly with tongs until the leaves are partially wilted before adding the next handful. Drain the spinach and let cool, then squeeze out as much liquid as possible. Coarsely chop the spinach and transfer to a large bowl. Return the saucepan to moderately high heat and repeat with the chard leaves, squeezing them dry.
2. Wipe out the saucepan. Add 2 tablespoons of the olive oil and heat until shimmering. Add the Swiss chard stems and ribs and cook over moderately high heat until crisp-tender, about 4 minutes. Add the scallions and sauté until tender, about 2 minutes. Add the scallions and Swiss chard stems and ribs to the bowl and stir in the feta, cottage cheese, parsley, dill, lemon juice and nutmeg. Season with salt and pepper and stir in the eggs.
3. Preheat the oven to 350°. In a small bowl, mix the clarified butter with the remaining 2 tablespoons of olive oil. Lightly grease a 9-by-13-inch glass or ceramic baking dish. Spread 1 sheet of phyllo dough on a work surface and brush with the butter mixture. Cover with a second sheet of phyllo and brush again. Continue until you have a stack of 8 buttered phyllo sheets. Repeat to form a second stack of 8 buttered phyllo sheets. Line the baking dish with a phyllo stack. Spread the spinach filling evenly on top. Cover with the second phyllo stack; trim the top layer of phyllo flush with that of the bottom layer. Roll the phyllo edges inward to form a border. Using a sharp knife, lightly score the top stack of phyllo to divide the pie into 8 even servings. Brush any remaining butter over the top.
4. Bake the pie in the bottom of the oven for 1 1/2 hours, or until the phyllo is golden and crisp. Let stand for at least 10 minutes before serving.
Make Ahead: The recipe can be prepared through Step 3 and refrigerated overnight.
Wine Recommendation: A round-textured Sauvignon Blanc will complement the pie filling's creamy texture. Try the 1998 Murphy-Goode Fumé Blanc Reserve from California or the 1999 Simunye from South Africa.
Notes: Clarifying butter separates the fat from the milk solids, which can scorch at higher temperatures. To make 1/3 cup of clarified butter, melt 1 stick (4 ounces) of unsalted butter in a small saucepan over moderately low heat. Let stand for 5 minutes, then skim off the foam. Pour the clear yellow liquid into a container and discard the milky residue. Clarified butter can be kept refrigerated or frozen for 2 months.