01 - Heat olive oil in a large pot over medium heat. Add diced onion and cook until softened, about 5 minutes.
02 - Add minced garlic and sauté for 1 minute until fragrant. Stir in tomato paste and cook for an additional minute.
03 - Add canned tomatoes with their juices, vegetable broth, sugar, salt, black pepper, and crushed red pepper flakes. Bring mixture to a simmer.
04 - Reduce heat and simmer uncovered for 20 minutes, stirring occasionally.
05 - Add fresh basil leaves. Use an immersion blender or transfer soup in batches to a blender; puree until smooth.
06 - Stir in heavy cream and adjust seasoning as needed. Keep warm over low heat.
07 - Butter one side of each bread slice.
08 - Place 4 slices of bread, buttered side down, on a skillet over medium heat. Layer each with 2 slices sharp cheddar and 1 slice mozzarella cheese.
09 - Top with remaining bread slices, buttered side up. Cook 2–3 minutes per side until golden brown and cheese is melted.
10 - Cut sandwiches in half and serve hot alongside the bisque, garnished with additional fresh basil.