Preheat the oven: Set the oven to 400°F to allow it to heat up while you prepare the pie.
Cook the scallions and chard stems: Place a 10-inch ovenproof skillet on medium heat and drizzle in 1 tablespoon of olive oil. Add the scallions, chard stems, and a small pinch of salt, cooking until they soften, which will take about 10 minutes.
Cook the greens: Add the remaining tablespoon of olive oil to the skillet and toss in the collard greens (or kale) in batches if needed to avoid overcrowding. Cook the greens for 3 to 7 minutes, stirring occasionally, until they wilt. Then, add the chard leaves and continue cooking until all the greens are tender, approximately 5 minutes more. Season with salt and pepper, then transfer everything to a large bowl and let it cool.
Mix the filling: In a separate bowl, combine the eggs, feta, cottage cheese, 1 tablespoon of melted butter, parsley, dill, and lemon zest. Stir until evenly mixed. Once the greens have cooled, add them to the mixture along with the lemon juice, and stir to combine. Taste and adjust seasoning with additional salt and lemon juice if needed.
Prepare the skillet and phyllo: Wash and dry the skillet thoroughly. Brush the inside (bottom and sides) with melted butter. Lay 2 sheets of phyllo dough in the skillet so that the edges hang over the sides. Then, add 2 more sheets of phyllo, placing them perpendicular to the first two sheets and again letting the edges hang over. Brush each sheet lightly with melted butter as you layer.
Fill and fold the phyllo: Spoon the greens mixture into the center of the phyllo layers, spreading it out evenly. Fold the overhanging edges of the phyllo over the filling, crumpling them slightly to create texture. Brush the top of the folded phyllo with the remaining melted butter to ensure it browns evenly during baking.
Bake: Place the skillet in the preheated oven and bake for 40 to 45 minutes, or until the phyllo is crisp and golden brown. Remove from the oven and let it cool slightly before serving. You can serve the pie warm or at room temperature, depending on your preference.