Preheat oven to 425 degrees F. Peel away the dry outer layers of skin from the garlic head, leaving skins and cloves intact. Cut off the pointed top portion (about 1/4 inch), leaving bulb intact but exposing the individual cloves. Place the garlic head, cut side up, in a custard cup. Drizzle with about 1/2 teaspoon of the olive oil. Cover with foil. Roast 25 to 35 minutes or until the cloves feel soft when pressed. Set aside just until cool enough to handle. Squeeze out the garlic paste from individual cloves.
Meanwhile, heat the remaining olive oil in a large skillet. Add mushrooms, carrots, parsnip, and onion. Cook 10 minutes, stirring occasionally. Add broth, thyme, salt, and pepper. Bring to boiling; reduce heat. Simmer, covered, 5 to 10 minutes or just until vegetables are tender. Stir in garlic paste.
In a medium bowl whisk together milk and flour until smooth; add all at once to mushroom mixture. Cook and stir over medium heat until thickened and bubbly. Stir in chicken. Divide mixture evenly among four 12-ounce individual casseroles. Set aside.
Unfold phyllo dough; remove one sheet of the phyllo dough and place on a flat surface. (As you work, cover the remaining phyllo dough with plastic wrap to prevent it from drying out.) Lightly coat the phyllo sheet with cooking spray. Lay another sheet of phyllo dough on top of the first sheet and lightly coat with cooking spray. Repeat with the remaining 2 phyllo sheets. Cut phyllo stack in half crosswise. Place one stack on top of the other to make a 9x7-inch stack of 8 sheets.
Using a pizza cutter, cut phyllo stack into 4 equal rectangles. Place a rectangle on top of the chicken mixture in each casserole.
Bake 15 to 20 minutes or until the filling is bubbly and phyllo is golden brown. Let stand 10 minutes before serving.