This study demonstrates that Large Language Models (LLMs) can transcribe historical handwritten documents with significantly higher accuracy than specialized Handwritten Text Recognition (HTR) software, while being faster and more cost-effective. We introduce an open-source software tool called Transcription Pearl that leverages these capabilities to automatically transcribe and correct batches of handwritten documents using commercially available multimodal LLMs from OpenAI, Anthropic, and Google. In tests on a diverse corpus of 18th/19th century English language handwritten documents, LLMs achieved Character Error Rates (CER) of 5.7 to 7% and Word Error Rates (WER) of 8.9 to 15.9%, improvements of 14% and 32% respectively over specialized state-of-the-art HTR software like Transkribus. Most significantly, when LLMs were then used to correct those transcriptions as well as texts generated by conventional HTR software, they achieved near-human levels of accuracy, that is CERs as low as 1.8% and WERs of 3.5%. The LLMs also completed these tasks 50 times faster and at approximately 1/50th the cost of proprietary HTR programs. These results demonstrate that when LLMs are incorporated into software tools like Transcription Pearl, they provide an accessible, fast, and highly accurate method for mass transcription of historical handwritten documents, significantly streamlining the digitization process.