Heat the oil in a large skillet over medium heat until shimmering.
Add the chicken and cook, breaking the meat into small pieces with a wooden spoon, until no longer pink, about 5 minutes.
Transfer to a strainer and let drain, about 5 minutes.
Break apart any large clumps of meat with your fingers.
Meanwhile, mix the feta, eggs, scallions, olives, mint, lemon juice, garlic, salt, pepper, and cayenne together in a large bowl.
Stir in the drained chicken until uniform.
(The filling can be refrigerated in an airtight container for up to 24 hours.).
For the phyllo layers:.
Adjust an oven rack to the middle position and heat the oven to 400 degrees.
Brush a 13 by 9-inch baking dish liberally with oil.
Lay 1 phyllo sheet in the bottom of the prepared dish and brush thoroughly with the oil.
Repeat with 7 more phyllo sheets, brushing each with oil.
Spread half of the chicken mixture evenly into the dish.
Cover with 4 more phyllo sheets, brushing each with oil.
Spread the remaining chicken mixture evenly into the dish.
Cover with 7 more phyllo sheets, brushing each with oil and sprinkling each with about 1 tablespoon of Parmesan.
Lay the final sheet of phyllo over the top and brush with oil (do not sprinkle the final layer with Parmesan).
Working from the center outward, use the palms of your hands to compress the layers and press out any air pockets.
Using a sharp knife, lightly score the pie into serving squares but do not cut through more than the top 3 sheets of phyllo.
Bake until the phyllo is golden and crisp, 30 to 35 minutes.
Cool on a wire rack for at least 10 minutes or up to 2 hours before serving.
(The pie can be wrapped tightly in plastic wrap and refrigerated for up to 4 days. Leftovers are best at room temperature or reheated in a 350-degree oven until crisp and heated through, about 25 minutes.).