01 - Heat olive oil in a large saucepan over medium heat. Add finely chopped onion and cook until translucent, about 5 minutes.
02 - Stir in minced garlic and cook until fragrant, approximately 1 minute.
03 - Add crushed tomatoes, vegetable broth, tomato paste, sugar, salt, and black pepper. Stir well and bring to a simmer.
04 - Reduce heat to low, cover, and simmer for 20 minutes, stirring occasionally.
05 - Remove from heat and stir in chopped basil. Purée the mixture using an immersion blender or countertop blender until smooth.
06 - Return soup to low heat, stir in the heavy cream, and warm gently. Adjust seasoning to taste and keep warm.
07 - Spread softened butter evenly on one side of each bread slice.
08 - Place 4 slices of bread, buttered side down, on a skillet over medium heat. Top each with 2 cheese slices and cover with the remaining bread slices, buttered side up.
09 - Cook sandwiches until golden brown and cheese is melted, approximately 3–4 minutes per side.
10 - Remove sandwiches from skillet and allow to cool for 1 minute. Slice each into 3–4 strips suitable for dipping.
11 - Ladle bisque into bowls, garnish with additional basil leaves, and serve alongside grilled cheese soldiers.