To make the Crust: Preheat oven to 350 degrees F.
Combine the ingredients and press into a buttered 9-inch pie pan. Spread the mixture evenly on bottom and sides of the pan and bake for 5 minutes. Remove the crust from the oven set aside and raise the oven temperature to 400 degrees F.
To make the Filling: Heat the butter in a large skillet over medium-high heat. Add the onion and mushrooms and cook, stirring, until lightly browned. Add the broccoli, chicken, salt, and pepper and stir to combine. Adjust the seasoning, as desired.
In a bowl, whisk together the eggs and milk. Add the vegetable mixture and cheese and mix well. Pour the mixture into the crust and bake for 15 minutes.
Reduce the oven temperature to 325 degrees F and bake until just set around the edges but slightly loose in the center, about 30 to 40 minutes. Let the filling set and cool before serving.