Preheat your oven to 350°F (180°C).
In a large roasting pan, combine the tomatoes, onions, garlic, and basil stalks. Add balsamic vinegar, sugar, a generous olive oil drizzle, and a salt sprinkle. Toss everything together and spread it out evenly in the pan.
Roast the mixture in the preheated oven for about 1 hour.
While the vegetables are roasting, prepare the basil oil. Set a bowl of ice-cold water near the stove. Bring a medium saucepan of water to a rolling boil and quickly blanch the basil leaves. Remove them with a slotted spoon and immediately transfer them to the ice-cold water to preserve their vibrant green color.
Squeeze out the excess water from the basil leaves and place them in a blender with olive oil and salt. Blend until smooth. Adjust seasoning to taste. You can use the oil as is or strain it through a sieve lined with kitchen paper to remove the basil pulp. Store the basil oil in a jar in the fridge for up to several weeks.
Once the vegetables have finished roasting, remove and discard the basil stalks. Transfer the tomatoes, onions, and garlic to a blender. Squeeze the roasted garlic flesh from the skins into the blender as well. Add enough water or stock to blend the mixture until completely smooth. You may need to do this in batches, depending on the size of your blender.
Strain the blended soup through a sieve into a large pot, pressing down on the pulp to extract as much liquid as possible. Bring the soup to a simmer and adjust seasoning and consistency, adding more liquid if needed. Simmer for about 10 minutes.
To serve, ladle the hot soup into bowls, top with basil oil, and sprinkle with parmesan or mozzarella cheese. Enjoy with crispy polenta chips for a true pizza-inspired experience.