Preheat the oven to 325°F (160°C). Cut the Beef Chuck into 2-inch cubes and pat dry; season generously with Salt and Black Pepper.
Heat a large heavy-bottomed pot or Dutch oven over medium-high heat and add a splash of Olive Oil. Cook the Smoked Bacon Lardons until crisp, then transfer them to a plate, leaving the rendered fat in the pot.
Working in batches so the meat browns rather than steams, sear the seasoned beef in the hot fat until well-browned on all sides; transfer browned pieces to the plate with the bacon.
Reduce the heat to medium and add a knob of Unsalted Butter to the pot. Add chopped Carrot and chopped Onion and cook until softened, about 6–8 minutes. Stir in minced Garlic for 1 minute.
Stir in the Tomato Paste and cook for 2 minutes to deepen its flavor, then sprinkle the All-Purpose Flour over the vegetables and stir to coat; cook 1–2 minutes to remove the raw flour taste.
Pour in the Red Burgundy Wine, scraping the browned bits from the bottom of the pot. Return the beef and bacon to the pot, add enough Beef Stock (and a splash of Water if needed) to almost cover the meat, then tuck in a Bay Leaf and a few sprigs of Thyme. Bring to a gentle simmer.
Cover the pot and transfer to the oven; braise until the beef is fork-tender, about 2 to 3 hours. Check once or twice and skim any excess fat from the surface.
While the beef braises, prepare the garnish: blanch the Pearl Onion briefly in boiling Water, slip off the skins, then sauté the peeled onions in a little Unsalted Butter and Olive Oil with a pinch of Sugar and a pinch of Salt until lightly caramelized and glazed. In a separate skillet, sauté the Button Mushrooms in Unsalted Butter and Olive Oil over high heat until browned; season with Salt and Black Pepper.
When the beef is tender, remove the pot from the oven, fish out and discard the Bay Leaf and thyme stems, taste and adjust seasoning with more Salt and Black Pepper if needed.
Stir the glazed pearl onions, sautéed mushrooms, and reserved bacon into the stew. Finish with a small knob of Unsalted Butter stirred in for gloss and sprinkle with chopped Parsley before serving.
Serve the Boeuf Bourguignon hot over mashed potatoes, buttered egg noodles, or crusty bread to soak up the sauce.
