Heat olive oil in a large pot or Dutch oven over medium heat.
Add the diced onion and sauté for about 3-4 minutes until softened.
Stir in the minced garlic and cook for an additional minute, until fragrant.
Add the ground beef (or turkey) to the pot and cook until browned, breaking it apart with a wooden spoon. Drain excess fat if necessary.
Add the diced carrot and bell pepper to the pot, stirring well to combine. Cook for an additional 3-4 minutes.
Stir in the chopped cabbage and cook down for about 5 minutes until it begins to soften.
Add the canned diced tomatoes (with juice), beef broth, tomato paste, oregano, thyme, paprika, salt, and pepper to the pot. Stir well.
Bring the soup to a boil, then reduce heat to low. Cover and simmer for about 30 minutes, allowing the flavors to meld.
If using, stir in the cooked rice just before serving, and heat through for another 5 minutes.