In a large stove-top pot, add a couple of glugs of olive oil and a pat of butter and bring to a sizzle over medium-high. Add the beef (cut into small 1/2 inch cubes). Do this in two batches, to not overcrowd the beef. Add a good pinch of kosher salt and some fresh pepper. Brown the beef cubes on all sides, stirring after a minute or two per side. We're not looking for perfection, just cook and stir until the cubes look mostly browned. Remove the beef and set it aside.
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 springs 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 and bring to a gentle boil. Cover and let simmer for approximately 1.5 to 2 hours, until the beef is tender and easily breaks apart with the 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 mushroom 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.