Handwritten Text Recognition (HTR) is essential for document analysis and digitization. However, handwritten data often contains user-identifiable information, such as unique handwriting styles and personal lexicon choices, which can compromise privacy and erode trust in AI services. Legislation like the ``right to be forgotten'' underscores the necessity for methods that can expunge sensitive information from trained models. Machine unlearning addresses this by selectively removing specific data from models without necessitating complete retraining. Yet, it frequently encounters a privacy-accuracy tradeoff, where safeguarding privacy leads to diminished model performance. In this paper, we introduce a novel two-stage unlearning strategy for a multi-head transformer-based HTR model, integrating pruning and random labeling. Our proposed method utilizes a writer classification head both as an indicator and a trigger for unlearning, while maintaining the efficacy of the recognition head. To our knowledge, this represents the first comprehensive exploration of machine unlearning within HTR tasks. We further employ Membership Inference Attacks (MIA) to evaluate the effectiveness of unlearning user-identifiable information. Extensive experiments demonstrate that our approach effectively preserves privacy while maintaining model accuracy, paving the way for new research directions in the document analysis community. Our code will be publicly available upon acceptance.