Heat the butter in a saucepan over medium heat and cook the onions for 4 minutes or until translucent.
Add the celery, carrots and thyme and cook for an additional 3 minutes.
Sprinkle 4 tablespoons of flour over the vegetable mixture. Stir to make a roux. Cook for 1-2 minutes, stirring to coat the vegetables.
Add the milk, broth, chopped turkey and peas and simmer over low to medium heat for 10-12 minutes or until thick and creamy. Add salt to taste.
Add the mixture to an 8 x 8 inch baking dish.
Top the turkey mixture with mashed potatoes and sprinkle shredded cheese on top to cover.
Bake for 25 minutes, until warmed through and cheese has melted.
Notes
To freeze: Cover and freeze after assembling the shepherd's pie. Make sure everything has cooled down before you put in the freezer. When ready to eat, you can defrost then bake as instructed, or bake from frozen and add 10-15 minutes to the cooking time.
Warm the Mashed Potatoes First: Cold mashed potatoes are hard to spread. Warm them slightly and loosen with a splash of milk if needed. For an even easier topping, transfer the potatoes to a piping bag or a zip-top bag with the corner snipped off and pipe them over the turkey mixture.
Thicken or Loosen the Filling: If the filling is too thin, whisk 1 tablespoon flour with a splash of broth and stir it in. If too thick, add a little extra milk or broth.