Preheat oven to 340 degrees F (170 degrees C).
Season Flat Iron Steak (2.5 pound) generously with Kosher Salt (to taste) and Freshly Ground Black Pepper (to taste). In a large Dutch oven or heavy ovenproof pot, heat the Olive Oil (2 tablespoon) over moderately high heat.
Add the Beef Stock (1 quart) and cook, turning occasionally until browned all over, about 12 minutes. Transfer beef to a plate; drain oil from pot and discard oil.
Return pot to moderately high heat and add Unsalted Butter (1/4 cup). Melt the butter, scraping up and mixing in the browned bits remaining in the pot.
Add the Carrot (2), Red Onion (1), Parsnip (1), Rutabaga (1), Celery (1 stalk), Cremini Mushroom (8 ounce), Garlic (8 clove), Fresh Rosemary (1 sprig), and Bay Leaf (2). Cook, stirring occasionally, until just tender, about 10 minutes.
Add Tomato Paste (3 tablespoon) and cook, stirring, for 4 minutes.
Add Dry Red Wine (1 1/2 cup) and boil until the liquid is reduced by half, about 8 minutes.
Return the beef and any accumulated juices to the pot. Add the stock or broth. Bring to a boil. Cover and transfer to the preheated oven. Braise until the beef is very tender, about 2 hours and 45 minutes.
Carefully remove the beef from the braising liquid to a separate dish and let cool slightly. Using a fork, break the beef into bite-size pieces and return to the pot. Season, if necessary, with Kosher Salt (to taste) and Freshly Ground Black Pepper (to taste).
Increase oven temperature to 400 degrees F (205 degrees C). Using the top of a 5-inch bowl as a guide, cut 6 rounds from the Puff Pastry (2 sheet). Transfer rounds to a parchment paper-lined baking sheet. Freeze until firm, about 15 minutes.
Brush pastry with the egg wash, made by beating Egg (1) and Water (1 teaspoon) together. Bake until the pastry is golden, about 10 to 12 minutes. Cool.
Ladle the hot stew into bowls and top each with a pastry round. Serve immediately.