Preheat the oven to 350 degrees . In a small skillet, melt 1 tablespoon butter over medium heat. Add the onion and cook until softened, about 5 minutes; transfer to a large bowl. Wipe out the skillet.
Place the spinach in a clean kitchen towel and squeeze out any excess water. Add the spinach, cottage cheese, feta, eggs and 1/2 teaspoon each salt and pepper to the onion and stir to combine. Set aside.
In the reserved skillet, melt the remaining 1 stick butter. Using a pastry brush, coat a 9-by-12-inch baking dish with melted butter.
Unfold the phyllo dough and line the baking dish with 1 phyllo sheet, gently pressing into place and letting any excess hang over the sides. Brush the sheet with more melted butter. Repeat until there are 8 layers of buttered phyllo in the dish. Cover the remaining phyllo sheets with a damp cloth to keep them from drying out.
Fill the phyllo-lined baking dish with the reserved spinach mixture. Fold in the overhanging phyllo, sheet by sheet, brushing butter on each sheet. Place another phyllo sheet over the filling and brush with butter. Repeat with the remaining phyllo sheets and butter, brushing the top layer generously with butter. Using a sharp knife, trim the overhanging phyllo sheets to fit the baking dish. Score the top to mark eight servings.
Bake the spanakopita until golden-brown, about 45 minutes. Let cool for 30 minutes before slicing.