01 - Heat 2 tablespoons of olive oil in a large saucepan over medium heat. Add the finely chopped yellow onion and cook until softened and translucent, approximately 5 minutes.
02 - Stir in the minced garlic and cook for 1 minute until fragrant.
03 - Add the crushed tomatoes, vegetable broth, tomato paste, sugar, 1/2 teaspoon salt, and 1/4 teaspoon black pepper to the saucepan. Stir to combine all ingredients thoroughly and bring the mixture to a gentle simmer.
04 - Reduce the heat to low, cover the saucepan, and allow the bisque to simmer for 20 minutes, stirring occasionally to prevent sticking.
05 - Remove the saucepan from the heat and stir in the chopped fresh basil. Use an immersion blender to purée the soup directly in the pan until completely smooth. Alternatively, carefully transfer the hot soup to a standard blender in batches and blend until smooth.
06 - Return the puréed soup to low heat and stir in the heavy cream. Warm gently without boiling, adjusting seasoning if needed with additional salt and pepper. Keep the bisque warm while preparing the grilled cheese.
07 - For the grilled cheese soldiers, spread the softened unsalted butter evenly on one side of each of the 8 bread slices.
08 - Place 4 slices of bread, buttered side down, onto a skillet or griddle preheated over medium heat. Top each of these slices with 2 slices of sharp cheddar cheese, then cover with the remaining 4 bread slices, buttered side facing up.
09 - Cook the sandwiches until the bread is golden brown and crispy, and the cheese is thoroughly melted, approximately 3 to 4 minutes per side.
10 - Remove the grilled sandwiches from the skillet and let them cool for 1 minute. Slice each sandwich into 3 to 4 uniform strips, known as 'soldiers'.
11 - Ladle the warm creamy tomato basil bisque into individual bowls. Garnish each serving with extra fresh basil leaves, and serve immediately with the grilled cheese soldiers for dipping.