Document understanding models are increasingly employed by companies to supplant humans in processing sensitive documents, such as invoices, tax notices, or even ID cards. However, the robustness of such models to privacy attacks remains vastly unexplored. This paper presents CDMI, the first reconstruction attack designed to extract sensitive fields from the training data of these models. We attack LayoutLM and BROS architectures, demonstrating that an adversary can perfectly reconstruct up to 4.1% of the fields of the documents used for fine-tuning, including some names, dates, and invoice amounts up to six-digit numbers. When our reconstruction attack is combined with a membership inference attack, our attack accuracy escalates to 22.5%. In addition, we introduce two new end-to-end metrics and evaluate our approach under various conditions: unimodal or bimodal data, LayoutLM or BROS backbones, four fine-tuning tasks, and two public datasets (FUNSD and SROIE). We also investigate the interplay between overfitting, predictive performance, and susceptibility to our attack. We conclude with a discussion on possible defenses against our attack and potential future research directions to construct robust document understanding models.