Preheat oven to 350°F. Line a baking sheet with parchment paper and set aside.
In a medium saucepan, heat oil over medium heat and add onions, cooking until they begin to soften (about 2 minutes), stirring occasionally.
Add flour and cook for 1 minute, stirring constantly.
Slowly add broth and milk, 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 (about 1-2 minutes), stirring occasionally.
Add chicken, mixed vegetables, 1 teaspoon thyme, and 1 teaspoon salt. Stir to combine. Decrease heat to medium and cook until warmed through, about 2-3 minutes. Remove from heat and set aside while you prepare the crusts.
Working with one Flatout at a time, spray each flatbread lightly with cooking spray on just one side. Place the flatbread, sprayed-side up, on prepared baking sheet and sprinkle 1/8 teaspoon dried thyme and 1/8 teaspoon salt evenly over cooking spray. Use your fingers to gently rub seasonings across flatbread to be sure they're evenly distributed. Flip flatbread over so the sprayed and seasoned side is down.
Spoon one-fourth of the prepared pot pie filling (about a heaping 1/2 cup of filling) onto one half of the Flatout, leaving a large border (as shown in photos in the post). Very slightly dampen the edges of the flatbread with a tiny bit of water, then fold the empty half of the flatbread over top of the filling, leaving about a 1" border where the bottom of the flatbread is not covered by the top half. Using a fork, press down gently but firmly all around the edge of the top half, to seal the edges. Then, fold the 1" border over onto the sealed top, and seal the hand pie a second time by again pressing a fork gently but firmly all around the edges.
Repeat this process with remaining 3 Flatout flatbreads and remaining pot pie filling, to make 4 hand pies total. Rearrange pot pies on parchment-lined baking sheet if needed, so they aren't crowded together, and bake for 20-22 minutes, until lightly golden-brown on the bottom and just lightly crispy outside but piping hot inside.
Serve immediately (careful – filling will be very hot), or refer to notes below for freezable, make-ahead directions.
Chicken: Nearly any boneless, skinless chicken breasts will work in this recipe, as long as they’re not overly flavorful from spices or marinades. 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 filling entirely ahead and store it in the refrigerator for a day or two, until you're ready to prepare the crusts and finish making these chicken pot pies.
Freezing: These hand pies freeze great after baking. To freeze, allow the pot pies to cool on the baking sheet for about 5 minutes, and then carefully move them to a cooling rack to finish cooling completely (to help prevent them from getting soggy on the bottom from sitting too long on the baking sheet). Once cooled, wrap them tightly in plastic wrap and store in freezer-safe storage bags. When ready to serve, remove individual hand pies from the freezer and unwrap, then bake (from frozen) at 350°F for about 20 minutes, until piping hot. Alternately, you can microwave each frozen hand pie for about 5 minutes at 50% power, but the crust will not be as crispy if you use this method of reheating.