Preheat large heavy bottom pot or Dutch oven on medium-high heat and swirl oil to coat. Add onion, garlic and carrots; cook for 3-4 minutes, stirring occasionally.
Add oregano and thyme and cook for 30 seconds, stirring often. Add ground beef, cook for 10 minutes, breaking into small pieces and stirring but allowing it to brown.
Add tomato paste and stir to incorporate.
Add diced tomatoes, tomato sauce, beef broth, cabbage, rice, vinegar, sugar, salt, pepper and bay leaves. Stir well, cover and bring to a boil. Then reduce heat to low and cook for 20 minutes for white rice or 35-40 minutes for brown rice. I recommend to stir a few times to prevent rice from sticking to the bottom.
Turn off heat and let soup stand for 10 minutes to allow flavors to meld. Add dill or parsley, stir and adjust any seasonings to taste, if necessary.
Serve hot topped with sour cream or Greek yogurt and a slice of bread.
Notes
Store: Refrigerate in an airtight container for up to 4 days.
Freeze: Fully cook, cool completely, and freeze in an airtight container for up to 3 months. Thaw in a refrigerator overnight.
Reheat: Reheat in a microwave or desired amount in a small pot on a stovetop until heated through, stirring a few times.