In a medium-size skillet, heat oil over medium-high heat. Add onion and cook until golden, about 4 minutes. Transfer to a large bowl.
Return skillet to the heat. Cook spinach, tossing, about 1 minute or until wilted. Drain spinach and let cool. Press out excess moisture and then roughly chop. Add spinach, beaten egg, feta, and cottage cheese to the bowl with onion and then season with salt and pepper.
Preheat oven to 350 degrees Fahrenheit.
Unroll one package of phyllo. Separate in half (about ten sheets each) and keep covered to prevent drying. Note: do not cover with a damp towel as advised in many recipes because the sheets are thin and delicate and will quickly turn into a mush. Cover them with foil, clear plastic wrap or dry towel. Work quickly to assemble.
Place one sheet on a work surface. Brush with oil. Lay another sheet on top and brush with oil. Repeat until you have 10 layers of phyllo. Spread half of filling lengthwise down the middle of phyllo stack. Roll into a log, brushing the seam with oil before sealing. Brush log with oil. Place the log, seam side down on a parchment-lined baking sheet.
Repeat with remaining phyllo and filling.
Using a small sharp knife, make 8-9 diagonal cuts on top of logs. Bake until deep golden brown 35-40 minutes.
Cut into slices along slashes. Serve warm or at room temperature.