Follow these steps to prepare your French Beef Stew perfectly:

1. Brown the Meat: In a large Dutch oven, heat olive oil over medium-high heat. Add the beef cubes, season with salt and pepper, and brown on all sides. Remove the beef and set aside.
2. Cook the Bacon: In the same pot, add the diced bacon or pancetta. Cook until crisp, about 5 minutes. Remove and set aside with the beef.
3. Sauté the Vegetables: Add the diced onion and sliced carrots to the pot. Sauté for about 5 to 7 minutes until softened. Add minced garlic and cook for another minute.
4. Incorporate Tomato Paste: Stir in the tomato paste and cook for another 2 minutes, allowing the flavors to combine.
5. Deglaze the Pot: Pour in the red wine and scrape up any browned bits from the bottom. Let it simmer for about 5 minutes.
6. Add Broth and Seasoning: Stir in the beef broth and bouquet garni. Return the browned beef and bacon to the pot. Bring to a simmer.
7. Slow Cook: Cover the pot and lower the heat. Let it simmer for 2 to 3 hours, stirring occasionally, until the beef is tender.
8. Add Mushrooms: About 30 minutes before the stew is done, add the quartered mushrooms to the pot.
9. Adjust Seasoning: Taste and adjust salt and pepper, if needed. Remove the bouquet garni before serving.

By following these steps, you'll create a French Beef Stew that's bursting with flavor, impressing everyone fortunate enough to savor it.