Heat oven to 400°F; press one crust into 9-inch deep-dish pie pan and set aside.
Melt butter in large skillet over medium; cook onion, carrots, and celery 5-7 minutes until softened.
Sprinkle in flour; stir constantly 1-2 minutes to remove raw taste.
Slowly add broth and milk, stirring constantly; simmer about 5 minutes until thickened.
Add turkey, peas, thyme, salt, and pepper; remove from heat and cool slightly.
Pour into crust; top with second crust and crimp sealed. Cut 4-5 steam vents.
Brush with egg; bake 30-35 minutes until crust golden and filling bubbling.
Cool 10 minutes before slicing; serve warm.