People may be able to join this study if they have a disease called eosinophilic esophagitis, or EoE, which can make it hard to swallow food. Before joining, they will have a screening visit and a procedure called an endoscopy with biopsies to see if they qualify. Participants will be placed into one of two groups by chance. One group will take a medicine called upadacitinib by mouth every day for 24 weeks. The other group will take a placebo, which has no active medicine, for 12 weeks and then switch to upadacitinib for 12 weeks. This allows everyone in the study to receive the study medicine at some point. During the study, participants will have regular checkups that include blood tests, vital sign measurements, questionnaires, and endoscopies with biopsies at the start of the study, at Week 12, and at Week 24. Researchers want to learn if upadacitinib can safely help reduce inflammation and improve symptoms in people with EoE.