Preheat oven to 400° F. Line a baking sheet with parchment paper. Grab 6 ramekins/cocottes to make mini chicken pot pies. To make a one-pot recipe, make sure you have a deep 12-inch pan.
Place your puff pastry on a floured surface. Roll out any cracks in the pastry with a rolling pin. Using your ramekins as a guide, cut out six rounds. (If you're making one-pot chicken pot pie, you can just leave it as a big rectangle). Place the puff pastry rounds/rectangle on a baking sheet, and brush with the egg wash. Bake for 12-15 minutes, until a deep golden brown. Set aside.
Place the chicken breasts (or thighs) and potatoes in the 12-inch pan, and cover with water. Bring to a boil, and reduce to a simmer. Cook 15-20 minutes, until the potatoes are fork-tender and the chicken is cooked through. Drain the chicken and potatoes. Then, cut the chicken in bite-sized pieces. Set aside.
Wipe the 12-inch pan clean, and place it back on the stove over medium-high heat. Melt the butter in the pan, and add the carrots, celery, and shallot. Cook, stirring occasionally until the vegetables are tender and starting to brown on the edges, about 8 minutes. Add the peas, corn, and garlic, and saute for about 30 seconds, until the garlic is fragrant.
Sprinkle the flour over the vegetable mixture, and cook for a minute or two. Then, slowly whisk in the chicken broth and milk, until the mixture is smooth and starts to thicken. Add the chicken, potatoes, thyme, parsley, salt, and lots of pepper. Taste, and season as necessary. Let the mixture simmer for about 10 minutes.
Divide the chicken pot pie filling between the six ramekins. Top with the puff pastry rounds, and serve! For the one-pot version, place the puff pastry rectangle on top of the filling. Cut into 6 slices, and serve. Enjoy!