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.

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.