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A common approach for testing fairness issues in text-based classifiers is through the use of counterfactuals: does the classifier output change if a sensitive attribute in the input is changed? Existing counterfactual generation methods typically rely on wordlists or templates, producing simple counterfactuals that don't take into account grammar, context, or subtle sensitive attribute references, and could miss issues that the wordlist creators had not considered. In this paper, we introduce a task for generating counterfactuals that overcomes these shortcomings, and demonstrate how large language models (LLMs) can be leveraged to make progress on this task. We show that this LLM-based method can produce complex counterfactuals that existing methods cannot, comparing the performance of various counterfactual generation methods on the Civil Comments dataset and showing their value in evaluating a toxicity classifier.