Preheat the oven: Preheat the oven to 350°F. You’ll lower the temperature once the stew goes into the oven.
Cook the bacon: Place a Dutch oven over medium heat and add the bacon. Cook until the fat renders and the bacon begins to crisp. Remove the bacon with a slotted spoon and set aside, leaving the bacon fat in the pot.
Sear the beef: Pat the beef cubes completely dry with paper towels and season with salt and pepper and lightly dust with flour. Increase the heat to medium-high and sear the beef in batches for 2–3 minutes per side, until deeply browned. Transfer the beef to a plate with the bacon.
Sauté the vegetables: Add the carrots and onion to the same pot and cook for about 5 minutes, until softened. Stir in the garlic and cook for 1 minute, then add the tomato paste and cook for another minute.
Deglaze with Cognac: Carefully add the Cognac and ignite with a long lighter or match, if desired. Once the flame subsides, scrape up the browned bits from the bottom of the pot.
Add the wine and herbs: Return the beef and bacon to the pot. Pour in the red wine and enough beef stock to nearly cover the meat. Add the thyme and bay leaf, season with salt and pepper, and bring to a gentle simmer.
Braise the stew: Cover the Dutch oven and transfer to the oven. Immediately lower the temperature to 325°F and braise for 2½–3 hours, or until the beef is fork-tender and the sauce has thickened.
Braise the mushrooms and onions: While the stew cooks, heat the butter and olive oil in a skillet over medium heat. Add the mushrooms and cook for about 10 minutes, until deeply browned and their moisture has cooked off. Add the pearl onions, red wine, thyme, bay leaf, and a pinch of salt and pepper. Simmer until the wine has reduced and the onions are tender.
Combine the stew: Remove the thyme stems and bay leaf from the beef stew. Stir in the mushrooms and pearl onions. If using leftover Beef Bourguignon, gently reheat it and add a splash of beef stock if the sauce has thickened too much.
Prepare the puff pastry: On a lightly floured surface, gently roll each sheet of puff pastry just until slightly thinner, about ⅛-inch thick. Don't over-roll or the pastry may become difficult to handle and won't puff as well.
Cut the pastry: Use your baking dishes as a guide to cut pastry rounds, leaving about 1 inch of overhang around the edges. Save the scraps for cheese straws or another puff pastry recipe.
Fill the baking dishes: Add the Beef Bourguignon to the baking dish, leaving a little room at the top for the pastry.
Top with pastry: Lay the puff pastry over the dish and gently press the edges to seal. Trim any excess pastry, as needed, then cut a few small vents in the center of the crust.
Brush with egg wash: Brush the puff pastry with egg wash, making sure to avoid pooling the egg wash around the edges. Sprinkle with a little Kosher salt.
Bake the pot pies: Place the baking dishes on a baking sheet and bake at 400°F for 20–25 minutes, or until the puff pastry is deeply golden and the filling is bubbling.
Cool and serve: Remove from the oven and let the pot pies rest for 5–10 minutes before serving. The filling will be very hot.