Brown the lardons. Place the lardons in a cold Dutch oven and set it over medium heat. Cook, stirring occasionally, until the fat has rendered and the edges are golden. Transfer the lardons to a plate with a slotted spoon, leaving the drippings in the pot.
Sear the beef in the drippings. Pat the beef dry and season with salt and pepper. Raise the heat to medium-high. Working in batches, arrange the cubes in a single layer and let a deep brown crust form before turning. Add a little olive oil between batches if the pot looks dry. As each batch browns, transfer it to the plate with the lardons.
Soften the vegetables. Lower the heat to medium. Add the onion and carrots and cook for about 5 minutes, stirring occasionally, until the onion softens. Stir in the garlic and tomato paste and cook for 1 minute more.
Return the meat and coat with flour. Add the beef, lardons and any juices from the plate back to the pot. Sprinkle the flour evenly over the meat and vegetables. Stir to coat and cook for 1–2 minutes, until the flour is incorporated.
Add the wine and stock. Pour in a splash of wine and scrape the browned bits from the bottom of the pot with a wooden spoon. Gradually stir in the remaining wine. Add the bay leaves and thyme, then pour in just enough stock to barely cover the beef and vegetables. Reserve any remaining stock.
Simmer gently for about 1 ½ to 2 hours. Bring to a simmer, cover and turn the heat to low. Look for an occasional bubble at the edge of the pot. Stir from time to time and begin checking the beef after 1½ hours. When a fork slides into a cube with only a little resistance, add the mushrooms. If the beef is still firm, cover and check again in 15–20 minutes.
Finish with the mushrooms. Stir the quartered mushrooms directly into the sauce. Simmer uncovered for 15–20 minutes, or until the mushrooms are tender, the beef pulls apart easily with a fork and the sauce lightly coats the back of a spoon.
Taste and serve. Remove the bay leaves and thyme stem. Taste before adding more salt, as the lardons and stock will have seasoned the sauce. Serve hot over Classic Buttery Mashed Potatoes.