Preheat the oven to 350°F. Lightly spray a 9-inch deep dish pie dish with cooking spray. Fit one pie crust into the dish leaving enough overhang to crimp with the top crust. Set aside.
In a medium-size skillet over medium-high heat melt the butter. Cook the green onion just until softened around 1-2 minutes.
Add the milk and cream cheese. Cook over medium-heat stirring until the cream cheese has melted., Add the garlic salt, black pepper and ground mustard.
Dissolve flour in 3 Tbsp cold milk until dissolved. If needed, add more milk 1 Tbsp at a time. Whisk the flour slurry into the the sauce.. Cook over medium-high stirring constantly until thickened. Once the sauce begins to thicken lower the heat and simmer for 2 minutes.
After 2 minutes mix-in 1 ½ cup shredded cheese. Stir until melted.
Remove from the heat and add the chopped chicken and broccoli florets. Mix until combined then pour into the pie shell. Sprinkle the top with the remaining ½ cup shredded cheese.
Fit the second crust on top. Fold and crimp the edges of the two pie crusts together. Cut 4 slits in the middle of the pie.
Whisk together the egg and cold water and brush over the the entire crust.
Bake for 40-45 minutes until golden. Rest for 10 minutes before serving.