Heat butter and olive oil together in a large pot over medium heat; saute onion and carrot until softened, 4 to 6 minutes. Add garlic, salt, pepper, and red pepper flakes; cook and stir until fragrant, about 2 minutes.
Stir crushed tomatoes, tomato juice, and diced tomatoes into onion mixture; bring to a simmer. Reduce heat, stir sugar into tomato mixture, and simmer until heated through, about 10 minutes. Add cream and simmer until heated through, 1 to 2 minutes. Add basil to soup and cook until wilted, about 3 minutes. Serve chunky or puree with an immersion blender.