Whisk egg in a large mixing bowl. Add chicken, bread crumbs, wing sauce, celery, blue cheese, and carrot. Mix well until evenly combined.
Form mixture into 6 patties and place on a baking sheet. Freeze until firm, about 30 minutes.
Heat oil in a large skillet over medium-high heat. Add fritters and cook, undisturbed, about 5 minutes. Carefully flip fritters and cook for 5 minutes more. Transfer patties to a paper towel-lined plate. Repeat with any remaining fritters.