Brown the Beef: Heat the olive oil in a large oven-safe pan over medium-high heat. Add the beef, seasoned with salt and pepper, and brown on all sides, about 2-4 minutes per side. Remove the beef and set aside.
Cook the Vegetables: Add the diced onion to the same pan and cook until tender, about 5 minutes. If using, add the shredded carrot and cook for another minute. Add the garlic and tomato paste, mixing and cooking for another minute.
Deglaze the Pan: Pour in the red wine (or beef broth) and scrape up the brown bits from the bottom of the pan using a wooden spoon.
Add the Broth and Spices: Add the beef broth, crushed tomatoes, cinnamon, allspice, nutmeg, bay leaves, and browned beef. Stir to combine.
Cook the Beef: Choose your cooking method:
Stovetop: Bring to a boil, then reduce the heat and simmer covered for 2-4 hours, until the beef is tender.
Oven: Cover the pan, transfer to a preheated oven at 275°F (140°C), and bake for 2-4 hours, until the beef is falling apart tender.
Slow Cooker: Transfer the mixture to a slow cooker and cook on LOW for 8-10 hours or on HIGH for 4-6 hours.
Add the Orzo: Once the beef is tender, add the orzo and cook:
Stovetop: Bring to a boil and cook for about 10 minutes, until the orzo is tender.
Oven: Cover and return to the oven to cook for another 10-15 minutes, until the orzo is cooked.
Slow Cooker: Cook on HIGH for 10-15 minutes, until the orzo is tender.
Serve: Remove the bay leaves, then serve the Beef Giouvetsi hot. Enjoy!