Heat your oven to 400°F so it's ready when the pie is. Dice the onion and slice the carrots and celery thinly. Shred the turkey and keep the pie crust close by so you can move quickly.
Melt the butter in your skillet over medium-high heat. Add the onion, carrots, celery, and thyme. Cook and stir now and then for 5 to 6 minutes, until the onion turns tender.
Sprinkle the flour over the vegetables and stir constantly for one minute. Pour in the broth and milk, then add the salt and pepper. Stir often until it simmers and thickens, then let it bubble gently for two more minutes.
Take the skillet off the heat. Fold in the shredded turkey and the frozen peas until everything is coated in sauce. Taste it, then add a little more salt and pepper if it needs it.
Roll the pie dough into a round about 10 inches wide. Lay it gently over the filling and tuck the edges in. Brush it with a thin layer of milk, then cut four 1-inch slits so steam can escape.
Slide the skillet into the hot oven for 20 to 25 minutes. You'll know it's ready when the filling bubbles at the slits and the crust turns deep golden. Keep an eye on it near the end so the top doesn't darken too much.
Let the pie cool for about five minutes on the counter. This short rest lets the filling thicken so it holds together on the plate. Then scoop it out and serve it hot.