In a large pot over medium-low heat, combine olive oil, onion, celery and garlic. Cover pot and cook, stirring occasionally, until vegetables are tender, 5 min. Raise heat to medium. Stir in broth, tomatoes, brown sugar and salt and simmer, 15 min. Use an immersion blender to purée slightly, leaving a few tomato chunks. (Alternately purée half the soup in a blender, holding lid on with a towel while blending.) Cover and return soup to a simmer.
Meanwhile, in a bowl, combine Bisquick, Parmesan and oregano. Add milk and stir until a dough forms. With Bisquick-dusted hands, roll dough into 12 balls. Add dumplings to soup; cover and simmer until dumplings are cooked through, about 10 min.
I typically grab my immersion blender and puree this soup right in the pot. No extra dishes to wash for me! But if you don't have one, ladle the soup into a blender and blend until smooth. Be sure to hold a kitchen towel over the blender; when blending hot liquids the steam can push the lid off causing a mess! I usually vent the center of the blender lid a bit and hold the towel over the vent.