Pour vegetable stock into a large pot; simmer over medium heat.
Melt half of the butter in a large saucepan over low heat; increase heat to medium. Add carrot; cook and stir until starting to soften, 3 to 5 minutes. Add onion, leek, celery and garlic; cook and stir until onion has softened, about 5 minutes. Add squash and courgette; cook for 2 minutes.
Add vegetable mixture into the vegetable stock; continue to simmer the soup over low heat.
Melt remaining butter in the saucepan. Cook and stir chanterelle mushrooms, oyster mushrooms, button mushrooms and portobello mushroom in the hot butter until starting to soften, about 5 minutes. Sprinkle flour over mushroom mixture; stir until a paste forms, 2 to 3 minutes.
Stir mushroom and flour mixture into the vegetable soup. Pour in cream; simmer until heated through, about 8 minutes. Season with cayenne, salt and pepper. Garnish soup with chives.