Preheat oven to 400 degrees F. Prepare a sheet pan with parchment paper or spray with nonstick spray.
Cube chicken into ½-inch pieces. Prepare the rest of the ingredients.
With the instant pot on the sauté setting, melt the butter. Add in the onion, carrot, celery, and garlic. Stir to combine and cook until softened, about 2 minutes.
Add in the chicken, potatoes, broth, thyme, salt, pepper, and celery seed. Stir to combine.
Pressure cook on high for 3 minutes with the valve in the "sealing" position. After the time is up, manually release the pressure by carefully switching the valve to the "venting" position.
Meanwhile, cut the puff pastry into 6 equal-sized squares. (I usually cut the pastry into 9 equal squares to have extra for snacking.) Add the pastry to the prepared sheet pan. Whisk the egg and brush it over the top of the pastry. Bake for 10 to 12 minutes, or until golden brown and slightly crispy. Set aside.
When the instant pot is done, remove about ¼ to ½ cup of just the broth from the pot. Add the flour to the broth and whisk the two together until there are no more clumps of flour. Set aside.
Add the frozen peas, milk, and parsley to the rest of the ingredients in the instant pot. Stir to combine. Add in the flour and broth mixture and mix again.
With the sauté setting turned on again, simmer to thicken the chicken pot pie filling, about 5 minutes, or until it reaches your desired thickness.
Ladle instant pot chicken pot pie into a bowl and top it with a puff pastry square. If you prefer, you can also cut the puff pastry in half widthwise to use half as the bottom of the "pie" and the other half as the top.