In a medium soup pot, melt the butter with the EVOO, 1 turn of the pan, over medium heat. Add the eggplant and garlic, season with salt and pepper and cook until softened, about 5 minutes. Add the chicken broth and tomatoes and bring to a boil. Add the raviolini and cook until tender, 3 to 4 minutes. Stir in the basil.
Serve the soup in bowls, topped with cheese.