First, be sure to thaw phyllo sheets per package directions. Once unwrapped, be sure to keep covered with a damp towel and/or plastic wrap to keep from drying out. Preheat oven to 375°.
In a large skillet, add the olive oil and the onion and cook, stirring, until softened, about 5 minutes.
Add the flour and cook, stirring constantly until the onion is coated with flour, about 1 minute. Gradually stir in the milk; cook, stirring constantly, until the mixture boils and thickens, 4-5 minutes. Add the mixed vegetables, cooked chicken, mushrooms, broth base, mustard powder, garlic powder, rosemary, salt and pepper. Return to a boil, stirring occasionally. Remove from heat and stir in the cheese.
Pour mixture into a 9" deep-dish pie plate or square baking pan. Make phyllo flowers: Cut sheets of phyllo into quarters. One piece at a time, grab the sheet from the middle bottom and pinch roughly 1/2" with fingers of one hand. With the other hand, pull the sheet upwards and twist into petal shape. Stick into top of chicken and vegetable mixture in whatever pattern you choose. Continue until top is covered or you are happy with how it looks. Don't worry if sheets tear, look uneven, or barely resemble petals of a flower! When you're happy with how it looks, give it a quick spray of olive oil or carefully brush with a bit of melted butter.
Slide pot pie into oven and bake for 30 minutes, or until bubbling at edges. Let stand about 5 minutes before serving. Prepare for ooh's and aah's when they see what you've created!