In a soup pot, heat the olive oil over medium heat until shimmering. Add the onion, carrot and parsley. Cook, stirring occasionally, until vegetables are softening, about 8 minutes.
Add the brown sugar and cook one minute more.
Add the salt, pepper, garlic powder, dried thyme, liquid smoke, crushed tomatoes and vegetable broth. Stir and bring to a simmer. Allow to simmer for about 20 minutes.
Remove the soup from the heat and blend until smooth using an immersion blender.
Return the soup to the heat and stir in the cream. Taste and adjust seasonings with salt if necessary.
Serve topped with fresh chopped parsley and grilled cheese sandwiches on the side.