To make the filling, heat the oil in a 12-inch cast iron over medium heat. If you plan to serve the pot pie in a casserole dish, any type of skillet will do. Add the mushrooms, carrots, celery, onions, and salt. Cook for about 10 minutes, while stirring occasionally, until the vegetables are mostly tender. Add the garlic, nutritional yeast, mustard, and poultry seasoning. Cook for a few minutes more until fragrant. Sprinkle the flour over the vegetables and stir to coat. Cook for about 3 minutes more then stir in the broth and the potatoes. Bring to a simmer and cook for about 10 minutes, stirring occasionally, until the potatoes are just turning tender and the sauce thickens. If the mixture gets too thick before the potatoes are tender, add a little more broth or water. When the potatoes are fork-tender, stir in the peas. Remove the skillet from the heat.

Unwrap the puff pastry but keep it on the sheet of paper. Use the rolling pin to roll the dough into a 12-inch square. Roll the pastry onto the pin, then unroll it over the skillet with the filling. Tuck the pastry into the skillet or use a sharp knife to trim any excess pastry and crimp it around the edge of the skillet. Cut several slits in a circular pattern in the center of the dough to allow steam to escape. Place the skillet on a baking tray lined with parchment to catch any spills. Bake for about 35 minutes on the lower rack of the oven. The crust should be golden brown when done. If you notice the edges browning too quickly, remove the tray from the oven, cover the edges loosely with foil and return it to the oven. Cool for about 15 minutes before serving.