Dice the onion, carrots, and celery into small, even pieces so they cook uniformly. Shred or dice your cooked turkey into bite-sized pieces. Set out your pie crust to take the chill off.

Melt the butter in a large skillet or Dutch oven over medium heat. Add the onion, carrots, and celery. Cook for 6-8 minutes, stirring occasionally, until the vegetables soften and the onions turn translucent. You want them tender but not mushy.

Sprinkle the flour over the vegetables and cook for 1 minute, stirring constantly to coat everything and cook out the raw flour taste. Slowly pour in the warm broth while whisking, then add the milk. Bring to a gentle simmer and cook for 3-4 minutes until the sauce thickens enough to coat the back of a spoon.

Stir in the turkey, frozen peas, and thyme. Season with salt and pepper - start with 0.5 tsp salt and adjust from there. Simmer for 2 minutes to heat everything through. The filling should be thick and creamy, not soupy. Remove from heat and let cool slightly while you prep the crust.

Preheat your oven to 400F. Pour the filling into a 9-inch deep-dish pie plate or leave it in your oven-safe skillet. Lay one pie crust over the filling, pressing gently to seal the edges. Trim any overhang to about 0.5 inch, then fold under and crimp with your fingers or a fork. Cut 4-5 slits in the top for steam to escape.

Brush the top crust with beaten egg for that golden sheen. Place on a baking sheet to catch drips and bake for 35-40 minutes until the crust is deep golden brown and the filling bubbles up through the vents. If the edges brown too quickly, cover them loosely with foil after 20 minutes. Let rest for 10 minutes before serving - this helps the filling set.