Brown beef. Heat a 5-quart Dutch oven or heavy soup pot over medium heat and add the olive oil (1 tablespoon). Add the ground beef (1 pound) and cook, breaking it up with a wooden spoon, until it is nicely browned and starting to develop deeper color in spots, about 8-10 minutes. Discard all but 2 tablespoons of drippings from the pot. If you don't have that amount remaining, add olive oil to make up the difference. Cook vegetables. Add the diced onion (1), green bell pepper (1), and red bell pepper (1) to the pot. Cook, stirring occasionally, until the vegetables are softened and the onion is translucent, about 5-7 minutes.
Add garlic and seasonings. Stir in the minced garlic (1 heaping tablespoon) and cook for about 30 seconds, just until fragrant. Add the fine salt (1/2 teaspoon), black pepper (1/2 teaspoon), sweet paprika (1/2 teaspoon), and Italian seasoning (1 teaspoon). Stir for about 30 seconds so the spices coat the meat and vegetables evenly.
Toast tomato base. Make a small well in the center of the pot and add the tomato paste (1 teaspoon). Cook, stirring it in place, for 30-60 seconds. Pour in the tomato puree (1 can) and stir well. Cook, stirring often, for 2-4 minutes, until the puree darkens slightly and loses its raw canned taste.
Simmer soup. Add the diced tomatoes (1 can), beef broth (4 cups), Better Than Bouillon beef base (1 heaping tablespoon), Worcestershire sauce (1 tablespoon), and brown sugar (2 teaspoons). Stir well until fully combined and the bouillon is dissolved. Bring the soup to a gentle boil, then reduce the heat to medium-low and simmer, uncovered or partially covered for 20-25 minutes, stirring occasionally, until the flavors have blended and the soup has thickened slightly.
Adjust seasoning and serve. Taste and adjust salt and pepper as needed. Spoon cooked rice (2 to 3 cups) into each bowl, then ladle the hot soup over the top. Finish with shredded mozzarella or a mozzarella-cheddar blend (1 cup) and chopped parsley, if desired.