Heat the oven to 180°c fan/200°c/Gas 6. Line a large roasting tin with baking paper and arrange the tomatoes in the tin cut-side up. Season with salt, then scatter the oregano, sugar and garlic. Drizzle with 1 tbsp of olive oil and 1 tbsp of vinegar. Roast for 45 minutes until the tomatoes have shrivelled and have released plenty of juices.
Meanwhile, set aside a large handful of basil leaves, reserving the stalks and remaining sprigs. Heat another 2 tbsp of olive oil in a saucepan and cook the onions with a little salt for 20 minutes until soft and sticky. Stir in the rest of the sugar and the remaining vinegar. Bring to the boil and simmer for 5 minutes until the onions have broken down and the liquid is syrupy.
When the tomatoes are done, tip them into the pan along with the juices, then pour in the vegetable stock and scatter the basil stalks. Stir well and bring to the boil. Reduce heat and simmer for 20 minutes until it becomes a chunky sauce.
Remove from heat and leave to cool slightly. Using a hand blender or food processor, blitz until smooth and creamy.
Place the reserved basil leaves in a food processor, with a pinch of salt and the remaining olive oil and mix to make a green oil.
Serve the soup in bowls and drizzle with basil oil.
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