Sauté carrots, parsnip, onions and garlic in a large sauce pan sprayed with non-stick spray heated on medium. Stir frequently until softened, about 5-7 minutes.
While veggies are cooking, stir in thyme, sage, poultry seasoning, salt and pepper.
Add flour and stir to incorporate and cook out the flour flavor, about 60 seconds.
Pour in chicken broth and use a wooden spoon to scrape up brown bits from the bottom of the pan. Bring to a boil and stir until thickened then reduce heat to low and simmer for 10 minutes.
Stir in green peas and shredded chicken. Check for seasoning and adjust with salt and pepper accordingly.
Pour chicken and vegetable mixture into a large baking dish.
Add olive oil to a small bowl. Working one sheet at a time, use a brush to "paint" olive oil on phyllo dough. After each layer is brushed with oil, add on top of the baking dish to form a top crust. Continue until all eight sheets are added making sure to paint the top of the last sheet with oil.
Use a knife to poke 2-3 slits in the phyllo crust then bake for 25-30 minutes or until phyllo is browned.