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Preheat oven to 375° F.
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Heat 2 tsp. oil in large nonstick skillet over medium heat.
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Add onion and carrots; cook, stirring frequently, for 4 to 5 minutes, or until onion is translucent.
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Add mushrooms; cook, stirring frequently, for 4 to 6 minutes, or until mushrooms are soft and most of the liquid has evaporated.
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Add 1 tsp. oil; cook, stirring frequently, for 1 minute.
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Add flour; cook, stirring frequently, for 1 to 2 minutes, or until flour and oil form a paste.
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Slowly add milk, broth, salt, pepper, and thyme. Bring to a boil, stirring frequently. Reduce heat to medium-low.
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Add peas, sweet potatoes, and chicken; cook, stirring frequently, for 2 minutes.
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Place chicken mixture in a 2½-quart baking dish. Set aside.
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Place phyllo on work surface. Keeping the original stack covered, remove one sheet at a time and quickly brush with remaining 3 tsp. oil. Cut oiled stack of phyllo into quarters. Place phyllo loosely on top of chicken mixture; covering evenly.
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Bake for 30 to 35 minutes, or until chicken mixture is bubbling and phyllo dough is golden brown.