In a large pot, heat the olive oil over medium heat. Add the chopped onion and sauté until it becomes translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the crushed tomatoes, vegetable broth, dried basil, and sugar to the pot. Season with salt and pepper. Bring to a simmer and cook for 15 minutes, allowing the flavors to meld.
Once the soup is done, blend it using a blender or immersion blender until smooth. If you prefer a creamier soup, stir in the heavy cream and heat through.
While the soup is simmering, heat a griddle or skillet over medium heat. Spread butter on one side of each bread slice.
Assemble the grilled cheese sandwiches by placing cheese between two slices of bread, buttered side out.
Cook the sandwiches on the griddle or skillet until golden brown, flipping carefully, about 3-4 minutes per side. Remove from the skillet and let cool slightly before slicing.
Serve the warm soup in bowls alongside the grilled cheese sandwiches, either whole or cut into halves.