Preheat the oven to 400°F, and drizzle a large sheet pan with 2 tablespoons of olive oil. If you don't have a large sheet pan, divide between two pans.
Place the halved tomatoes cut side up on the sheet pan, along with the quartered onion and smashed garlic.
Drizzle with the remaining 1 tablespoon of olive oil. Sprinkle the vegetables with the brown sugar, salt, and pepper.
Roast for 45 to 65 minutes, or until everything is soft, and the tomatoes are caramelized in spots, and bubbly on top. Note: If you're using two pans, reduce the roasting time to 45 minutes, rotating the pans halfway through the cooking time.
Carefully transfer the roasted tomatoes, onion, garlic, any juices, and 1 tablespoon of tomato paste to a blender. Blend until smooth.
To remove the seeds and skins, set a strainer over a sauce pot and pour the blended tomato mixture through the strainer.
Set the pot over medium heat. Stir in 1 cup of broth and add more if needed to reach your preferred consistency. Bring the soup to a gentle simmer to warm the broth through.
Give the soup a taste and season with more salt and pepper if needed.
Remove the pot from the heat and stir in the basil and cream (or milk), if using and serve. Garnish with extra fresh basil.