Cook Aromatics: Heat the olive oil in a large pot over medium heat. Add onion and sauté until soft, about 5 minutes. Stir in garlic and cook until fragrant.
Simmer Soup: Add tomatoes, stock, sugar, salt, and pepper. Bring to a gentle simmer, then reduce heat and cook 12–15 minutes.
Puree & Finish: Remove from heat and blend until smooth. Return to low heat, stir in cream, adjust seasoning, and keep warm.
Assemble Sandwiches: Butter (or mayo-spread) one side of each bread slice. Layer mozzarella and cheddar on the unbuttered side, then top with a second slice (buttered side out).
Grill: Over medium–low heat, cook sandwiches 3–4 minutes per side, pressing lightly, until golden and cheese is melted.
Serve: Slice sandwiches diagonally. Ladle soup into bowls, garnish with basil, and serve hot alongside your grilled cheese.