In a large pot or Dutch oven, melt butter over medium heat. Add onion, shallots, and garlic and cook until softened, about 5 minutes.
Add carrots, celery, and bell pepper and cook for another 5 minutes, until slightly tender.
Stir in sun-dried tomatoes, tomato paste, lime juice, sugar, salt, pepper, and cayenne pepper. Cook for 1 minute, stirring constantly.
Pour in chicken broth and add diced tomatoes. Bring to a boil, then reduce heat and simmer for 15 minutes.
While the soup simmers, prepare the dumplings. In a bowl, combine flour, Parmesan cheese, basil, baking powder, and salt. Whisk to combine.
Add the egg and stir until a dough forms. If the dough is too dry, add a teaspoon of water at a time until it comes together.
Bring a pot of water to a boil. Drop small spoonfuls of dumpling dough into the boiling water and cook for 3-5 minutes, or until they float to the surface and are cooked through.
Serve the soup hot, topped with the Parmesan dumplings and a sprinkle of fresh basil.