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.
Heat the oil in a large skillet over medium heat. Saute the onions until soft and caramelized, about 25 minutes. Season with salt and pepper. Transfer the onions to a plate.
Generously butter both sides of each bread slice. In between two bread slices, pile the caramelized onions between two layers of cheese.
Heat a large skillet or griddle over medium-low heat. Cook the sandwiches, flipping once, until the cheese is melted and the bread is golden brown, 2 to 3 minutes per side.