CLIP is a powerful and widely used tool for understanding images in the context of natural language descriptions to perform nuanced tasks. However, it does not offer application-specific fine-grained and structured understanding, due to its generic nature. In this work, we aim to adapt CLIP for fine-grained and structured -- in the form of tabular data -- visual understanding of museum exhibits. To facilitate such understanding we (a) collect, curate, and benchmark a dataset of 200K+ image-table pairs, and (b) develop a method that allows predicting tabular outputs for input images. Our dataset is the first of its kind in the public domain. At the same time, the proposed method is novel in leveraging CLIP's powerful representations for fine-grained and tabular understanding. The proposed method (MUZE) learns to map CLIP's image embeddings to the tabular structure by means of a proposed transformer-based parsing network (parseNet). More specifically, parseNet enables prediction of missing attribute values while integrating context from known attribute-value pairs for an input image. We show that this leads to significant improvement in accuracy. Through exhaustive experiments, we show the effectiveness of the proposed method on fine-grained and structured understanding of museum exhibits, by achieving encouraging results in a newly established benchmark. Our dataset and source-code can be found at: https://github.com/insait-institute/MUZE