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.