In a large skillet over medium heat, heat 2 tablespoons of oil. Add the onions and cook, stirring frequently, until they start to soften, about 5 minutes.
Add the garlic and dill and cook with the onions, stirring frequently, until fragrant, about 2 minutes.
Add the spinach, salt and pepper to the skillet in two batches and cook with the onion mixture until well combined, 5-7 minutes total.
Remove from heat and stir in the green onions and lemon zest until evenly combined.
In a medium bowl, whisk the eggs, then add the feta cheese and continue to whisk until well combined. Stir into the spinach mixture.
Preheat the oven to 350°F and lightly oil a 9x13-inch baking dish with 1 tablespoon of oil.
Unroll one sleeve of phyllo sheets and cut off an inch at the end to make them fit into the 9 x 13 pan.
Lay half a sleeve (10 phyllo sheets) in the prepared pan. Spread the spinach mixture over the phyllo in one even layer. Lay the other half sleeve of 10 phyllo sheets over the spinach mixture.
Drizzle the remaining 3 tablespoons of oil over the top layer of phyllo sheets and use a pastry brush to brush the oil all over. Cut the spanakopita into 12 pieces.
Bake until golden brown all the way into the cut lines, 45-50 minutes, rotating the spanakopita halfway through baking.