Heat olive oil in a large pot over medium heat. Add the chopped onion and cook for 3–4 minutes until soft and translucent. Stir in the garlic and cook for 1 minute until fragrant. Avoid letting the garlic brown, or it’ll turn bitter.

Add the tomatoes, vegetable broth, and sugar to the pot. Bring to a gentle boil, then reduce the heat to low. Cover and simmer for 20 minutes. If using fresh tomatoes, they should break down completely.

Use an immersion blender to purée the soup until smooth (or carefully transfer to a blender in batches). Stir in the fresh basil and heavy cream. Season with salt and pepper to taste. Keep the soup warm on low heat.

Butter one side of each sourdough slice. Heat a skillet over medium heat. Place one slice butter-side down, sprinkle with cheese, then top with the second slice (butter-side up). Cook for 3–4 minutes per side until golden and crispy. Let cool slightly, then cut into bite-sized cubes.