In a large nonstick skillet, heat oil over medium heat and add onion, cooking until the onion begins to soften (about 2 minutes), stirring occasionally.
Whisk flour into milk. Slowly add milk mixture and broth to pan, whisking continuously as you do so (to avoid lumps), until the mixture is smoothly combined. Increase heat to bring mixture to a boil. Once boiling, cook until slightly thickened to a gravy-like consistency (about 4-5 minutes), stirring occasionally. (You can cook it until it's slightly thicker, or slightly thinner, depending on how thick you personally want it.)
Reduce heat to medium. Add turkey, mixed vegetables, thyme, and salt. Stir to combine. Cook until vegetables are hot, about 5 minutes.
Meanwhile, toast 4 slices of bread. Remove pot pie mixture from heat and serve immediately over toast, garnishing with a little fresh parsley (if desired for a pretty presentation).
Broth: We used fat-free, reduced-sodium chicken broth in testing this recipe. It's hard to find reduced-sodium turkey broth, so if you'd like to use turkey broth instead of reduced-sodium chicken broth, you'll likely need to reduce the amount of kosher salt added to the recipe, to compensate for the higher level of salt in the turkey broth you're using. I recommend, in this case, that you wait to add the salt until the end of cooking, and that you add the salt a little at a time, tasting as you go to be sure you don't over-salt.
Turkey (or chicken): You can make this recipe with either leftover turkey or chicken. When we make it with chicken, we usually grab a store-bought rotisserie chicken, which is a quick and easy option, but you can also simply plan on cooking a couple of extra chicken breasts whenever you’re making chicken, so you have extra cooked chicken for this recipe without any extra work.
Make-Ahead Steps: You can make the pot pie "filling" entirely ahead and store it in the refrigerator for a day or two, until you're ready to toast the bread and serve your pot pies. Simply reheat the "filling" gently while the bread toasts, and it's ready to go.
Freezing: The "filling" also freezes well for make-ahead meal prep.