The National Museum of Iraq, located in Baghdad, is a cultural hub that documents the ancient history of Mesopotamia. While the museum primarily focuses on artifacts and the rich heritage of the region, its context also involves the culinary history as part of its cultural tapestry. Mesopotamian cuisine has influenced the Middle Eastern culinary traditions with its array of spices, grains, vegetables, and meats (excluding cow meat due to your restriction). This includes a variety of breads, legumes, and stews that have been staples for centuries. The museum itself doesn't directly exhibit food, but the history of the Iraqi cuisine can be indirectly explored through the agricultural practices and domestic life depicted in the ancient artifacts displayed.
Please note that the addresses provided are basic and might require further specific details for exact locations. Also, when traveling to Iraq, it is essential to ensure current travel safety advisories and the availability of services at these destinations.