Preheat oven to 375°.
Melt butter in a large Dutch oven over medium-high heat; add oil. Add potatoes and next 6 ingredients (through thyme), and sauté for 5 minutes. Reduce heat to medium-low; sprinkle flour over vegetables. Cook for 5 minutes, stirring frequently. Stir in milk and broth. Increase heat to medium-high; bring to a boil. Reduce heat, and simmer 5 minutes or until thickened. Add chicken, peas, salt, and pepper.
Spoon mixture into a 3-quart baking dish. Place 1 phyllo sheet on a large cutting board or work surface (cover remaining dough to keep from drying); lightly spray with cooking spray. Repeat layers with cooking spray and remaining phyllo. Place phyllo layers loosely on top of mixture in dish. Place dish on a baking sheet. Bake at 375° for 30 minutes or until top is golden.