Measure out 1/4 cup water into a glass measuring cup and stick it in the freezer.
In a large bowl, whisk together 2 cups flour, sugar, and 1 teaspoon salt.
Use a pastry cutter or fork to cut in the butter and shortening. Cut until it looks like coarse crumbs. Don't overdo it! You want pea-size pieces of butter.
In a small bowl beat one egg. Pour half of the egg (you can just eyeball it) into another small bowl and save it for brushing the top of the pie.
Add the 1/4 cup water in the freezer to the half beaten egg.
Beat it together, then add it to the shortening mixture. Stir until it is just coming together, then use your hands to knead it once or twice (only to make it come together). Divide the dough in half.
Generously sprinkle flour onto a sheet of parchment paper or a pastry cloth (or right on the counter) Roll the dough with a rolling pin, using as much flour as you need so that it doesn't stick. Invert the parchment paper onto a deep 9-inch pie pan and gently peel off the paper. (Or transfer it the way that sounds easiest to you. I often loosely roll the dough onto my rolling pin and then unroll it into the pie plate.) Stick the pie plate in the fridge if you have room.
Roll out the remaining dough and set aside.
Preheat your oven to 425 degrees F.
Chop the onion, carrots, and celery.
In a large skillet, melt 1/2 cup butter over medium heat. When it is hot, add the onions, carrots, and celery. Saute, stirring occasionally until veggies are soft, about 5-7 minutes.
Make the roux. Tilt the pan* to one side so the butter pools. Add the flour to the butter (try to keep the veggies separate to avoid flour lumps) and whisk together.
Still tilting the pan, gradually (this means not all at once!) pour in the chicken broth and milk. Heat should still be on medium. Whisk together, still avoiding the veggies so that you don't have to deal with flour lumps. When all the lumps are gone, stir in the veggies.
Add 3/4 teaspoon salt, pepper, celery seed, and the sprig of thyme, if you're using it.
Cook the sauce for a couple minutes until it thickens a bit.
Add the chicken and peas and stir. Remove from heat. Remove the thyme stem.
Pour the mixture into the pie shell.
Top with the other pie shell. Fold the top pie shell underneath the bottom so that it is sealed, then crimp however you like.
Brush the top of the pie and the crust with the remaining half beaten egg. Use a sharp knife to make several slits in the top of the pie so that it can vent.
Bake at 425 for 15 minutes. If you are worried that the pie will spill over, place it on a baking sheet.
After 15 minutes, cover the crust of your pie so it doesn't get too brown.**
Continue baking for another 12-15 minutes, until the crust is golden brown all over.
Remove from the oven, let stand 10-15 minutes, then devour.