Preheat oven to 375°F.
Heat a non-stick pan over medium heat. Once hot, add the beef, carrots, celery, onion, salt, garlic, and thyme leaves. Cook, breaking the beef up into smaller pieces using a spatula until it is no longer pink.
Add the flour. Stir well, until no dry flour remains. Slowly stir in the milk and broth. Cook until bubbling and thick, stirring often, about 5-7 minutes.
Remove from the heat. Stir in the peas. Remove the thyme sprigs. Let this cool for 10 minutes before filling the pie.
Keep the pie crust in the tin it comes with and place it onto a baking sheet so that it is safer and easier to move it in and out of the oven.
Using a slotted spoon, scoop the meat and vegetables into the pie crust. Add some of the remaining thickened liquid to the pie crust (you don't want the filling to be overly soupy so add just enough until it is nice and moist).
Place a pie crust top or another pie crust over the top. If using another pie crust, trim any excess crust around the edges and seal the edges together, crimping with a fork. Cut 3-4 slits in the top crust to allow steam to escape.
Bake for 30 minutes on the bottom rack of the oven, or until golden brown.
Cool for 15-20 minutes before portioning and serving.