Melt butter in a large pot over medium heat. Add diced onion and cook until soft and translucent, about 5 minutes.
Add minced garlic and cook for another minute until fragrant.
Pour in crushed tomatoes, broth, and sugar. Stir well and bring to a simmer. Let it cook for 15 minutes.
Use an immersion blender to puree the soup until smooth, or carefully transfer to a regular blender in batches.
Stir in heavy cream and season with salt and pepper. Heat through but don't boil.
While soup simmers, make grilled cheese sandwiches on a griddle or in a skillet until golden brown on both sides.
Serve soup hot with grilled cheese on the side for dipping!