Beat the eggs and combine with the feta cheese. Toss with the chopped spinach in a bowl.
Heat the olive oil in a skillet until it shimmers. Add the onions and garlic and saute until the garlic is aromatic, then add the lemon juice. Pour the pan contents over the spinach mixture. Add the chopped oregano and dill. Toss to combine all the ingredients completely.
Coat the inside of a glass baking dish (about 11 by 7 inches) with butter, then melt the remaining butter. Cut a piece of phyllo big enough to cover the bottom and overlap the sides of the dish. Brush the phyllo with butter, then place another sheet of phyllo over it. Butter and repeat until you have four sheets of phyllo in the dish.
Spoon the spinach mixture into the baking dish. Fold the overlapping phyllo on top of the spinach. Cut the remaining two sheets of phyllo to fit exactly, with butter between the layers. Make six or so cuts into the pastry to let steam out.
Bake the pie in the 350-degree oven until the phyllo browns and the mixture is heated all the way through--25-30 minutes. Allow to cool to just warm. Cut into squares and serve.