Preheat the oven to 400 degrees F. Cut a small piece off of each end of the acorn squash so it sits upright on either side. Then cut in half crosswise.
Scoop out the seeds and set each one face down on a rack set over a sheet tray. Place in the oven and pour 1/4 inch of water into the pan. Cook for 30 minutes until al dente, remove from oven and let cool.
Add oil to a sauté pan over medium-high heat. Combine the mushrooms, chicken, onion and peas in a saute pan and sprinkle with salt and pepper to help release the juices.
Add the flour and stir to cook through for about 2 minutes. Add the warmed chicken broth and stir until thickened, about 3 minutes.
Turn off the heat and let cool slightly. Taste mixture and adjust seasoning, adding salt and pepper as needed.
Preheat the oven to 350 degrees F. Cut the filo dough into circles 2-inches wider than the circumference of your squash. Use a small plate as a stencil if helpful.
Spoon the pot pie mixture into the acorn squash.
Cover with filo dough circles and brush with melted butter.
Place in the oven for 30 minutes, until golden brown on top. Remove and let cool slightly before serving.