Heat the oil in a large saucepan over medium heat. Add the garlic and onions and cook until the onions are translucent, about 5 minutes. Add the brown sugar and tomatoes and cook for 5 minutes more. Add 1/2 cup water, the thyme and Parmesan rind, if using. Cook for about 25 minutes.
Discard the thyme sprigs and Parmesan rind. Puree the soup in a food processor or with an immersion blender until smooth. Strain through a fine sieve set over cleaned saucepan. Stir in the cream and butter. Season with salt and pepper.
To serve, ladle the soup into bowls and sprinkle with Parmesan and chives.
Serve with Grilled Cheese with Caramelized Onions.