Heat 2 tablespoons olive oil in a heavy large pot over medium heat. Add onions and cook, stirring occasionally, until very tender, about 25 minutes. Add minced garlic and cook, stirring, for 1 minute. Add chicken broth, tomatoes, and 6 tablespoons of grated Parmesan cheese. Simmer soup uncovered 20 to 30 minutes, stirring occasionally. Season, to taste, with salt and pepper.
Ladle soup into bowls. Sprinkle with thinly sliced basil and remaining Parmesan.