Preheat your oven to 370 degrees F and grease a 9-inch round baking pan. Set aside.
Heat up the oil in a large pan or pot over medium-high heat. Add the carrots, celery, and onions then sprinkle with 1/2 teaspoon salt. Cook for 3-4 minutes until the veggies get a bit brown and soften, stirring often.
Add the chicken, garlic, and 1 teaspoon of salt. Cook for another 3-4 minutes until the chicken is not pink anymore. (The chicken will finish cooking in the oven later.) Add butter and stir until it has melted completely. Now add in the flour and stir thoroughly until everything is coated.
Pour in the chicken broth and heavy cream. Stir until the sauce gets creamy. Bring to a boil (keep stirring!) and then reduce the heat to medium-low. Let the filling simmer for 2-3 minutes until the sauce is thick and creamy.
Stir in frozen peas, nutmeg, and thyme. Then season with salt and pepper to taste. Take the pot off the heat and set it aside.
Place one sheet of pie crust carefully into the greased pan and cut off any excess dough. (Be sure to leave yourself a bit of dough to crimp though!) Pour in pot pie filling.
Top the pie with the second sheet of pie crust. Seal the edges flute them with your fingers or crimp them together with a fork. Cut some slits into the top of the crust to let the steam out. Brush the beaten egg over the top crust and sprinkle with poppy seeds if desired.
Bake on the lowest oven rack for 40-45 minutes until the crust is golden brown. (If the edges start to get too dark, add some tin foil on top to prevent them from burning.)
Remove the pot pie from the oven and let it rest for at least 15 minutes before cutting into it. This helps the filling to set and not be too soupy when served.
Notes
You can add other vegetables like sliced mushrooms, chopped green beans, or small broccoli florets to the filling.
Instead of using fresh chicken, you can substitute shredded pre-cooked or rotisserie chicken, or use some leftover turkey. If you use pre-cooked chicken, just omit adding the chicken in step 4 and add it later when you add in the peas.