Melt 3 tablespoons of butter in a stockpot over medium heat.
Add 1 cup of chopped onion, 1/2 cup of chopped carrot, 2 chopped celery stalks, 2 minced garlic cloves, and 1/4 cup of chopped flat-leaf parsley.
Cook and stir until the onion becomes soft and translucent, about 5 minutes.
Add 2 1/2 cups of chopped tomatoes, 1 can of tomato sauce, 3/4 cup of brewed coffee, 1/4 cup of water, 1 teaspoon of white sugar, 1 teaspoon of salt, and ground black pepper to taste.
Bring the soup to a boil, then reduce heat and simmer for 20 to 25 minutes.
Carefully pour the hot soup into a blender, filling the pitcher no more than halfway full.
Hold down the blender lid with a folded kitchen towel and use quick pulses to start blending the soup.
Continue blending until the soup is smooth, then pour it into a clean pot.
Stir in 1/3 cup of heavy cream and heat until warm, but do not boil.