Melt butter in a stockpot over medium heat. Cook and stir leeks and garlic in melted butter until tender, about 5 minutes.

Stir chicken stock, carrots, celery, curry powder, turmeric, ginger, black pepper, and red pepper flakes with the leeks and garlic; bring to a boil, reduce heat to medium-low, cover the pot, and simmer the mixture until vegetables are tender, about 30 minutes.

Pour coconut milk into the soup and stir; cook just until hot, 1 to 2 minutes.