To peel the tomatoes, dip in boiling water for 30 seconds or until skins start to split. Dip in cold water. When cool enough to handle, remove skins and core tomatoes. Coarsely chop; set aside.
In a 4-quart Dutch oven heat butter over medium heat until melted. Add garlic; cook and stir for 30 seconds. Add onion; cook for 4 to 5 minutes or until tender. Add chopped tomatoes. Bring to boiling; reduce heat. Simmer, covered, for 30 minutes, stirring occasionally.
Transfer tomato mixture in batches to a food processor or food mill. Cover and process until smooth. (If using a food mill, discard tomato seeds and onion.) Return pureed mixture to Dutch oven. Meanwhile, finely chop basil; reserve 1/4 cup.
Stir the remaining basil, salt, and pepper into tomato mixture; heat through. Add cheese, cream, and, if desired, Armagnac. Cook and stir just until cheese is melted.
Ladle soup into bowls. Sprinkle with reserved chopped basil.