1. Preheat oven to 400°F (200°C). Set out an 8x11x2-inch baking dish and a baking sheet that's large enough to hold the dish.
2. Heat a large, heavy-bottomed pot (like a Dutch oven) over medium heat. Once it's hot, pour in ½ tablespoon of the olive oil. Add half of the mushrooms to the pot and let them cook undisturbed for a full 2 minutes. Stir them up and season with salt and pepper. Let them cook undisturbed for another 2 minutes. Once they start browning and glistening, and they've shrunk down in size slightly, transfer the sautéed mushrooms to a medium bowl.
3. Pour another ½ tablespoon of olive oil into the pot and repeat the sautéing process with the remaining mushrooms. As you get towards the end of cooking this batch, pour in a couple tablespoons of the wine and quickly scrape up all the brown bits with your spoon. Transfer the second batch of mushrooms to the medium bowl.
4. Give the pot a little rinse and wipe it out before returning to the stove over medium heat.
5. Pour in the remaining olive oil and swirl it around. Add the shallots and carrots to the pot. Season with salt and pepper. Cook the shallots and carrots, stirring occasionally, for about 6-7 minutes, or until edges of carrots have softened slightly.
6. Add the garlic, thyme, paprika, and bay leaves to the pot and cook for another minute.
7. Add the tomato paste to the pot and cook for another minute.
8. Pour in a splash of the red wine and use your spoon to scrape up any brown bits. Then, add the sautéed mushrooms, remaining red wine, and all but 2 tablespoons of the vegetable stock. Give the bourguignon a good stir and then bring it up to a hard boil for two minutes. Lower the heat to a simmer and let it cook, uncovered, until the liquid has reduced by a third, about 25 minutes. Turn off the heat.
9. Stir the arrowroot flour into the remaining vegetable stock. Add this slurry to the bourguignon and stir for about 30 seconds. Stir in the Tamari too. Check the bourguignon for seasoning and adjust if necessary. It should thicken up considerably! Remove the bay leaves and discard them. Transfer the bourguignon to the baking dish and gently spread it out to the edges in one even layer. Place the baking dish on top of the baking sheet. Let the bourguignon cool in the baking dish for at least 15 minutes.
10. Sprinkle a bit of flour on a clean work surface. Roll the puff pastry out until it's an inch bigger than your baking dish on all sides. Loosely roll the pastry onto the rolling pin to transfer it over to your baking dish. Then, unroll it out over the bourguignon in the dish. Let the sides overhang--they'll puff and shrink up.
11. Using a sharp knife, cut three vents into the center of the puff pastry. Then, brush the pastry with the non-dairy milk. Sprinkle the top with coarse salt and pepper if you like.
12. Bake the pot pie for 25 minutes, or until pastry is golden brown and filling is bubbling. Let the pot pie cool for 5-8 minutes or so before digging in and serving. This allows the stew portion to thicken back up a bit. Enjoy warm with sides of your choice.