Heat butter and olive oil together in a deep large skillet over medium heat; cook and stir mushrooms and shallots until slightly softened, about 5 minutes. Add cherry tomatoes and cook until tomatoes begin to cook down, about 5 minutes.

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Reduce heat to medium-low and add cream to mushroom mixture. Stir zucchini noodles and basil into mushroom mixture; cook and stir until heated through and zucchini are slightly tender, about 5 minutes. Season mixture with salt and pepper.