Preheat the oven to 350 degrees. Thaw 6 sheets of phyllo dough overnight in the refrigerator or one hour before using. Bring the chicken stock to a simmer and keep warm.
In a 6-quart pot, add the olive oil and flour, and cook until lightly golden, about 3 minutes. Add the onion, garlic, leek, carrot, celery, red potato and turnips, and stir with a wooden spoon, making sure the bottom does not get too dry. Add the chicken, bay leaf and thyme, and continue stirring for about 3 minutes, so that everything is coated with the flour mixture.
Add the warm chicken stock, stir and scrape the bottom of the pot to get all the color into the mixture and keep it from scorching. Bring to a boil, then lower to a simmer. Let the mixture simmer, covered, for another 5 to 7 minutes, until it becomes thick and the chicken is opaque. Remove from heat and reserve.
Meanwhile, take 6 sheets of phyllo dough and, with a knife, cut out a square slightly larger than the rim of a 9-x-9-inch cake pan and brush with some olive oil. Set aside.
. Remove the bay leaf and thyme sprigs from the chicken mixture and transfer the chicken to the cake pan. Top with the phyllo dough and press it down firmly with your hands. Place the pie on a baking sheet and bake in the oven at 350 degrees for 30 minutes, turning the tray around halfway through the cooking time. Remove from the oven, let cool for 10 minutes, cut into six portions and serve.