Cut the tomatoes in half, place them cut-side down in a foil-lined grill pan and drizzle with some oil. Grill for 4–5 minutes until the skins have split and blackened. Meanwhile, fry the onion, potato and garlic in the remaining oil for 5 minutes, stirring occasionally until softened and turning golden around the edges.
Peel and roughly chop the tomatoes and add to the onion and potato with the pan juices, then stir in the stock, tomato purée, sugar and vinegar. Add half the basil, season and bring to the boil. Cover and simmer for 15 minutes.
Purée half the soup in batches in a blender or food processor until smooth. Return to the saucepan with the rest of the soup and reheat. Season to taste, then ladle into bowls, garnish with the remaining basil leaves and serve with Parmesan twists.