Preheat oven to 400°F. Heat 2 tablespoons of the oil in a large Dutch oven over medium-high. Add greens in batches, letting them wilt slightly after each addition before adding more, about 4 minutes. Cook, stirring often, until greens are softened and have released their water, 3 to 5 minutes. Scrape into a colander in kitchen sink; let cool for 15 minutes. Wipe Dutch oven clean.

While greens cool, lightly grease a 9- x 13-inch baking dish with ½ teaspoon of the oil. Remove phyllo sheets from package; cover phyllo sheets with a very lightly dampened paper towel to prevent drying out. Place 1 of the phyllo sheets in an even layer in prepared baking dish; lightly brush with ½ teaspoon of the oil. Lay another phyllo sheet on top, and brush with ½ teaspoon of the oil. Repeat process with 4 of the phyllo sheets and 2 teaspoons of the oil for a total of 6 layers.

Bake in preheated oven until phyllo is golden brown and crisp, 10 to 12 minutes; set aside.

Heat 2 tablespoons of the oil in same Dutch oven over medium. Add onion, garlic, pepper, and salt, and cook, stirring often, until onion softens, about 6 minutes. Remove from heat.

Working in batches, place cooled greens on a clean kitchen towel, and squeeze firmly over sink to remove as much liquid as possible. Transfer greens to bowl with onion mixture, and stir to combine.

Whisk together eggs, lemon zest and juice, and nutmeg, if using, in a medium bowl until thoroughly blended. Stir in scallions and herbs. Stir egg mixture into greens mixture until thoroughly combined; gently fold in feta and provolone until just combined.

Spread filling evenly over cooked phyllo. Brush 1 of the phyllo sheets with ½ teaspoon of the oil; gently crumple phyllo sheet, and place on top of filling. Repeat process with remaining 5 phyllo sheets and 2 ½ teaspoons oil, placing phyllo sheets on top of filling forming 2 rows of 3.

Bake at 400°F until phyllo is golden brown and crisp, 10 to 12 minutes. Let cool for 10 minutes before serving.