Preheat oven to 400°F. Grease a 2-quart casserole dish or deep-dish pie plate; set aside (or you can use a 2-quart oven-proof skillet).
In a separate bowl, whisk together chicken broth and half-and-half. Set aside.
Melt butter in a large skillet over medium-high heat. Add onion, carrot, and celery; sauté for 10 minutes. Add flour and continue cooking and stirring for 1 more minute. Reduce heat to medium and stir in the chicken broth mixture.
Continue cooking over medium heat, stirring regularly, until thick (about 5-6 minutes). Make sure that the mixture doesn’t boil. Stir in the salt and pepper. Add chicken and frozen peas.
Transfer the mixture to the prepared baking dish (or leave in the oven-safe skillet). Top with the pie crust, sealing and fluting the edges as desired. Cut slits into the top of the crust to allow the steam to escape.
Bake in a 400°F oven for about 35-40 minutes, or until the crust is golden brown.
Let the pot pie rest for 10-15 minutes before serving.