Pre-heat oven to 350°F. Grease a 9-inch pie plate with butter or nonstick spray. Place one of the pie doughs on top. Set aside.
Melt the butter in a large skillet on medium heat. Cook the onion until it starts to soften (about 2 minutes). Cook for 1 more minute.
Mix in the flour, and cook for 1 minute. Slowly add the milk and then the cream cheese. Stir frequently until cream cheese is melted and mixture thickens. Season the mixture with salt, pepper and mustard.
Turn off the heat and add 1 1/2 cups cheddar cheese to the saucepan. Stir until smooth. Add the chicken and broccoli.
Mixture in a pie shell. Top with the remaining cheese. Place the second pie crust on top. Fold and crimp the edges to seal. Four slits should be made in the middle of the pie.
Mix egg with 1 tablespoon water, and then brush the top crust.
Bake for 40 to 45 minutes. Allow to cool for 10 minutes before you serve. Enjoy!