Preheat oven to 350°F. Butter a 13x9x2-inch baking dish or spray with nonstick spray.
In a medium skillet, heat the olive oil over medium heat for about 1 minute. Add the onions and cook until softened, about 5 minutes, stirring often. Set aside.
In a large bowl, stir together onion, drained spinach, ricotta, feta, jack cheese, eggs, dill, parsley, salt and black pepper. Spread half of filling in baking dish. Cover with 2 pieces of the phyllo pastry (just tuck the edges in on the sides). Brush pastry lightly with melted butter. Top with one piece of phyllo. Tuck in any overhanging edges. Spread remaining filling over phyllo. Top with 2 more pieces of phyllo and brush lightly with butter. Top with remaining phyllo sheet and brush with butter. Use a sharp knife to mark 12 squares by cutting through the top layers of pastry.
Bake until phyllo topping is golden and filling is bubbling gently, about 45 minutes. Use a sharp knife to cut through the marked squares. Serve warm.
Notes
*If covered tightly and refrigerated, this pie can be assembled early in the day and baked later. Cold pot pie will probably need to bake an additional 5 minutes.