1. Heat butter and olive oil in a medium saucepan over medium heat until butter is melted, about 1 minute. Add onions and carrots and cook, stirring occasionally, until lightly caramelized and softened, about 5 minutes. Add corn and continue to cook until corn is lightly caramelized, 7 to 8 minutes. Add shallots and garlic and cook until fragrant, about 1 minute.

2. Add corn stock, salt, cayenne, and crab boil to saucepan. Bring to a boil and immediately reduce heat to a simmer. Cook soup until corn is very tender, 25 to 30 minutes. Remove from heat and let cool slightly.

3. Working in batches, transfer soup to the jar of a blender and puree until smooth. Strain soup through a fine mesh sieve into a clean saucepan. Add cream and crawfish tails and return to a boil. Remove from heat and let cool completely.

4. Transfer soup to refrigerator until chilled. To serve, ladle soup into chilled bowls; garnish each with 1 tablespoon creme fraiche and 2 teaspoons chives.