Heat the oil over high heat in a 12-inch heavy-bottomed pan that has a a tight fitting lid. Generously salt and pepper the beef, then add about 1/3 of the beef cubes to the hot oil. Do not move the beef cubes right away, but let them sear well on the first side, about 3 minutes.
Toss in the smashed garlic cloves in their peels. Use tongs to turn the beef cubes and sear each of the other sides before transferring to a rimmed plate. Add more oil if needed and repeat the searing process with the next third and then the final third of the beef. Return all of the beef to the pan, still over high heat.
Add 1 cup of the Chianti to the pan and let it boil, scraping the bottom of the pan to loosen any of the stuck flavourful bits. Let the wine boil for about 2 minutes, or until it has reduced slightly. Add the beef broth or stock, petite diced tomatoes, and tomato paste to the pan and stir to combine. Nestle the thyme bouquet or dried thyme and bay leaves into the sauce, and return to a boil.
As soon as it boils, add the lid, drop the heat as low as it can go, and let it simmer for 2 hours, or until the beef is tender. Remove the lid, stir in the remaining 2/3 cup of Chianti and the freshly ground black pepper, raise the heat to medium, and simmer for about 20 to 30 minutes, or until the sauce is quite thick and a deep mahogany colour. Remove the thyme bouquet (if you used it), the garlic peels, and the bay leaves. This can be served immediately but improves immensely when allowed to rest tightly covered in the refrigerator for 24 hours before serving.