Eosinophils play a key role in the pathogenesis of eosinophilic oesophagitis. Therapies that suppress eosinophil recruitment and activation may give a benefit. Mepolizumab is a humanised monoclonal antibody against interleukin-5 (IL-5). This study will evaluate the ability of mepolizumab to decrease the recruitment and infiltration of eosinophils into the oesophagus, thereby reducing the inflammation and symptoms of EE (eosinophilic oesophagitis) in adult patients.