In a large saucepan, heat oil over medium heat; add in onion, shallot, and garlic; cook about 5 minutes or until onion is tender, stir often.
Add in chicken broth, carrot, celery, red pepper, lemon juice, and sugar; bring to a boil; decrease heat, cover and simmer for 20-25 minutes or until vegetables are very tender.
Cool slightly; do not drain.
Carefully transfer half or the mixture to a food processor or blender; cover and process or blend until smooth; repeat with the remaining mixture (or can use stick blender to puree soup in the pot).
Return all pureed mixture to the same saucepan; stir in undrained tomatoes and cayenne pepper (to taste); cook, uncovered, over low heat about 10 minutes or until heated through, stirring often.
Meanwhile, fill a Dutch oven half full with water; bring water to a boil.
Make the Dumplings--in a small bowl, combine flour, fresh basil, baking powder, and salt; in another small bowl, stir together the beaten egg and vegetable oil; pour egg mixture into flour mixture; using a wooden spoon, beat until a soft, sticky dough forms.
Drop Basil Dumplings dough from a slightly rounded 1/2 teaspoon measuring spoon into boiling water; cook for 4-5 minutes (start timing after dough rises to the surface) or until dumplings are cooked through, turning once.
To test for doneness, remove a dumpling and check the center, making sure it is cooked through.
Using a slotted spoon, remove dumplings from Dutch oven; drain in a colander; rinse under cold water; drain again.
Add half of the dumplings to soup in pan; heat through.
Season soup to taste with salt and pepper; ladle into soup bowls; top with remaining dumplings; garnish with basil sprigs.