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Preheat the oven to 200 °C (about 400 °F). brush a 9" x 13" sheet pan, with olive oil, to the top edges. Whisk the egg and olive oil, then add the yogurt and milk and whisk again until smooth. Set aside. Mix the cheeses and set aside. Place two sheets of filo in the sheet pan, so it completely covers the bottom and sides, leave 1 in. hanging over the edges of the on each side and brush with melted butter. Spread a couple of tablespoons of the egg mixture on the filo. Tear 2 sheets of filo into 6-7 pieces, as well as the excess you trimmed off. Dip each piece into the egg mixture and layer it, on top, in a wavy shape, so all of the base is covered. Each piece of filo should be moist, but not excessively wet. Scatter the cheese mixture evenly over and top with flat leaf parsley. Tear 1 sheet of filo into large pieces again and dip into the egg mixture layering them in a wavy shape, just like the previous layer. You should be able to cover all of the filling with this. Now place a full rectangular piece of filo on top, and lightly brush with melted butter---repeat with 1 more sheet, then fold in the sides over the edges, making sure to brush this last sheet with the egg mixture on top. Don’t over do it, there maybe some mixture leftover. For an extra crispy top, drizzle a tablespoon or so of olive oil on top. Bake in the middle of the oven until the börek is golden on top and cooked through, about 30-40 minutes. Cool, then cut into 10-12 pieces. Enjoy hot, lukewarm or cooled.