Preheat the oven to 350 degrees F (175 degrees C).
Arrange tomatoes cut-side up on 1 or 2 large 4-sided baking pans. Place unpeeled garlic on top of tomatoes. Drizzle with olive oil and sprinkle with coarse salt.
Roast in the preheated oven for 1 hour. Remove from the oven and let cool in the pan.
Meanwhile, heat butter in a large stockpot over medium-low heat. Cook onion, sugar, and oregano in the hot butter until onion is softened, 5 to 7 minutes. Peel roasted garlic and add to the pot. Add roasted tomatoes and their juices. Pour in vegetable stock. Cover and let soup simmer for 20 minutes.
Puree soup using an electric blender working in batches, using extreme caution with hot liquids; vent the lid a bit and cover with a old towel. Force puree through a mesh strainer into another pot, discarding solids.
Stir in heavy cream and season soup with salt and pepper.
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