Pour vegetable broth into a large pot; bring to a simmer over medium heat.
Melt 1/4 cup 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 is translucent, about 5 minutes. Add yellow squash and zucchini; cook for 2 minutes.
Pour carrot mixture into the vegetable broth. Keep broth simmering over low heat.
Melt remaining 1/4 cup 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 broth. Pour in heavy cream; simmer until heated through, about 8 minutes. Season with cayenne, salt, and pepper. Garnish soup with chives.