Preheat oven to 375°F.
Lightly brush 2 (9") round baking pans with oil. Set aside.
Place 1 phyllo sheet on a clean work surface, keeping remaining sheets covered with a damp towel (spritz it, don't soak it) to prevent them from drying out.
Brush phyllo sheet lightly with 1 teaspoon oil. Top with a second phyllo sheet. Brush top of second sheet with 1 teaspoon oil.
Place 2 oz of meat mixture in a thin line along 1 long side of phyllo sheet. Starting at filling end, roll phyllo (snug, but not too tight or it will tear) around filling to form a long log shape. Coil log to form a tight spiral. Set it in the center of one of the 9" pans.
Repeat process, but only up to rolling phyllo around filling to form a long log shape. Bring it to the pan, and continue to form the coil where the first one left off, and keep going repeating the process until you fill the pan with what looks like one large coil. Then do it again in the second pan.
Whisk 1 tbsp olive oil with the egg, and brush it onto the börek.
Bake at 375°F for 25-30 minutes until golden brown. You can move to the second shelf under the broiler to help browning for the last 1-2 minutes.
Let cool for 3-4 minutes, then remove from pans and slice into triangles like a pizza and serve.