Prepare Baking Dish - About 1 - 2 hours before serving, begin to assemble and bake the borek. Lightly brush the bottom and sides of a 9x13 baking dish with 2 tsp of the melted butter. Trim phyllo sheets to fit evenly with rim of the dish.
Prepare Oven - Set one rack on the uppermost rung and another rack on the lowest rung. Preheat oven to 425℉ (220℃).
Prepare Filling - In a food processor, pulse cheeses to small pieces or grate. Remove blade and mix in the parsley, pepper, and salt.
Assemble Borek - In the medium bowl with remaining butter, combine the milk and beaten egg with the butter. Line the bottom of the baking dish with half the phyllo by laying down 1 sheet and brushing each sheet evenly with the milk mixture. Repeat with the remaining 5 sheets. Spread the cheese mixture over the surface of the stacked phyllo sheets. Top with the remaining six sheets, brushing all but the top sheet with the milk mixture. Cut pie into 2-inch squares with a sharp knife.
Top Borek - Add the egg yolk and flour to the remaining milk mixture. Stir with a soup spoon to make a light batter. Spread evenly over the top layer and rest for 15 minutes.
Cook Borek - Set the pie on the bottom rack of the oven and lower heat to 350℉ (175℃). Bake for 15 minutes. Transfer pie to the top rack and continue baking until the top becomes golden brown and all the liquid has been absorbed, about 30 minutes. Remove from the oven to a baking rack and let cool for 5 minutes. Loosen the squares and serve hot.