Render pork belly over medium-high heat until fat releases from the pieces and the pork belly becomes crispy. Remove from the pot, leave the fat in.
Toss the beef cubes in flour. Shake off excess flour. Working in batches, sear beef in the pot until deeply browned on all sides. You may need to add some oil or reserved fat. Set aside.
After removing the beef, lower the heat in the pot to medium. Add the onions and carrots to the pot, saute until softened. Add tomato paste, cook until it becomes brick red. Add garlic, saute for 30 seconds.
Preheat the oven to 325ºF.
Remove the pot from heat (especially if cooking on a gas stove). Add the Cognac (or brandy). Simmer for 2-3 minutes, or until the alcohol smell softens to a rounder and slightly sweet aroma.
Empty the bottle of wine into the pot. Scrape up the fond on the bottom. Let the wine reduce a bit over medium heat for 5-10 minutes. Return the beef and lardons to the pot. Add thyme, bay leaf, and enough beef stock to barely cover the meat. Cover with the lid. Put the pot in the oven for 2 ½ to 3 hours, until the sauce has thickened and the beef becomes tender.
When the braise is done, check the sauce. If the sauce looks thin, remove beef and reduce on stovetop until it is nappe and you can drag your spoon through it and it leaves a trail that closes slowly. Return beef.
While the beef is braising, prepare the garniture by sauteeing the mushrooms in butter until they become golden brown. Set aside.
Bring a saucepan of water to a rapid boil. Blanch the pearl onions for 60 seconds. Shock them. Pat dry. In a pan, glaze them by combining butter, sugar, and water, simmering for 10 minutes.
Add mushrooms and onions to the braised beef.
Finish with a knob of cold butter, a splash of red wine vinegar, and some grated chocolate to round out the mouthfeel of the sauce.
Serve with pommes puree, butter noodles or crusty bread. Top with chopped parsley.