Melt the butter in a large pot over medium heat. Add the onion and carrot and cook for 8 minutes, stirring now and then, until soft. Add the garlic and cook for 30 seconds, then stir in the tomato paste and cook for 1 to 2 minutes, until it darkens slightly.

Add the tomatoes with their juices, crushing them against the side of the pot with a spoon. Stir in the broth, sugar, salt and pepper. Bring to a boil, then reduce the heat and simmer, partly covered, for 20 minutes, until the carrot is very tender.

Meanwhile, make the croutons: butter one side of each bread slice. Make 2 sandwiches with the cheese inside and the buttered sides out. Cook in a skillet over medium-low heat for 3 to 4 minutes per side, until golden and the cheese has melted. Let rest for 2 minutes, then cut into 1-inch squares.

Remove the soup from the heat and add the basil. Blend with an immersion blender until smooth, or blend in batches in a countertop blender with the lid vented and covered with a towel. Stir in the cream and warm gently without boiling. Taste and add more salt or a pinch more sugar if the soup tastes sharp.

Ladle into bowls and top with the grilled cheese croutons, a drizzle of cream and torn basil leaves.