Preheat oven to 340°F. Season beef generously with salt & pepper. In a large Dutch oven or heavy ovenproof pot, heat the oil over moderately high heat. Add the beef 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 butter. Melt the butter, scraping up and mixing in the browned bits remaining in the pot. Add the carrots, onion, parsnip, rutabaga, celery, mushrooms, garlic, rosemary, and bay leaves. Cook, stirring occasionally, until just tender, about 10 minutes. Add tomato paste and cook, stirring, for 4 minutes. Add wine 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 salt & pepper.

Increase oven temperature to 400°. Using the top of a 5-inch bowl as a guide, cut 6 rounds from the pastry. Transfer rounds to a parchment paper-lined baking sheet. Freeze until firm, about 15 minutes. Brush pastry with the egg wash. 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.