Place beef pieces in a medium mixing bowl that has a tight-fitting lid. Sprinkle flour, salt, and pepper evenly over the beef. Cover bowl tightly, then flip and shake the bowl a few times to completely coat the beef. If you do not have a bowl with a lid, place these ingredients in a ziplock bag and then flip and shake to coat the beef. Set aside.

If using a slow cooker with a brown/sauté function: Heat slow cooker and add 1 tablespoon of the oil and the tablespoon of butter. Once hot, add diced onion and carrot. Sauté, stirring occasionally, until vegetables are just softened, about 6 minutes. Stir in garlic and sauté 1 more minute. Remove vegetables to a large plate or a rimmed pan. If you do not have a brown/sauté function on your slow cooker: Complete this step in a large nonstick skillet on the stovetop over medium-high heat.

If using a slow cooker with a brown/sauté function: Add 2 more tablespoons of the oil to the hot slow cooker, still on the brown/sauté function. Once oil is shimmering hot, add half the beef in an even layer. Once the bottom sides of the beef pieces are very nicely browned (which means, added flavor!), flip them over and brown opposite sides. I use a tongs for flipping. Then remove beef to the plate with sauteéd vegetables. Add remaining 1 tablespoon of oil and let it come to shimmering hot. Then add remaining half of beef in an even layer, and repeat the browning process. Remove beef to the plate with other beef and vegetables. If you do not have a brown/sauté function on your slow cooker: Complete this step in a large nonstick skillet on the stovetop over medium-high heat.

If using a slow cooker with a brown/sauté function: Turn slow cooker to high heat and add wine, scraping and stirring any remaining bits and juices from bottom of slow cooker. Bring wine to a boil, and then cook for 2 minutes. If you do not have a brown/sauté function on your slow cooker: Complete this step in the same large skillet used to cook the vegetables and beef, scraping and stirring any remaining bits and juices into the wine.

If using a slow cooker with a brown/sauté function: Add beef and vegetables back to slow cooker. Then add diced tomatoes (do not drain), tomato paste, beef broth, thyme, oregano, bay leaves, carrot coins, and mushrooms. Cover and cook on high for 3 hours. Then taste test to see if stew needs more salt and/or pepper. With the ingredients I use, I usually add an additional 2 teaspoons of kosher salt and 1/2 teaspoon of black pepper at this point. Cover and cook for 1 more hour, to allow the flavors to deepen and the beef to get "fork-tender". If the beef is not ultra tender after 4 hours, let it cook a bit longer until it is. If you do not have a brown/sauté function on your slow cooker: Carefully pour hot wine into slow cooker, and turn slow cooker to high heat. Follow remaining instructions from this Step 5, until beef is fork-tender.

Discard bay leaves. To serve, ladle hot stew into individual bowls. Garnish with fresh thyme, oregano, and small basil leaves, if desired.