Heat the olive oil and butter in a large Dutch oven or heavy-bottomed pot over medium-high heat. Working in batches, brown the beef on all sides (a minute or two per side). Add a good pinch of kosher salt and some fresh pepper. **Don't overcrowd the pot or the beef will steam rather than brown. Transfer the browned beef to a plate and continue with the remaining beef. To the same pot, add an additional glug of olive oil and add the diced onions and shallots. Cook on medium-high, stirring occasionally, for 2-3 minutes. Add in minced garlic and let cook for an additional minute.
Add the wine and deglaze the pot while scraping all the browned bits from the beef and onions into the wine. Bring the wine to a boil, sprinkle the flour over the top, and whisk thoroughly until well incorporated. Let boil for 5 minutes.
Add the beef broth, tomato paste, three sprigs of fresh thyme, bay leaf, and carrots. Return the beef to the pot, plus a couple of pinches of kosher salt and fresh pepper. Stir all to combine. Bring the stew to a gentle simmer, then reduce the heat to low. Cover and let it simmer gently for 1½–2 hours, or until the beef is tender and easily pierced with a fork.
15 minutes before serving the bourguignon, add the quartered mushrooms to a very hot stove-top pan (ready with olive oil and a pat of butter). Let the mushrooms brown in place for a few minutes before stirring. Add 2 sprigs of thyme to the pan and sprinkle with salt and pepper. Cook on medium-high, stirring occasionally until the mushrooms are browned, nearly all over.
Stir the mushrooms into the beef bourguignon. Serve with rustic bread or spoon over creamy mashed potatoes.