I don't know much about Eosinophilic Esophagitis, but shortly after my celiac disease diagnosis I kept a food diary on an Excel spreadsheet (you could use Google Sheets) to track what I ate, and what my reactions were, if any. This allowed me to identify foods that caused me IBS-like symptoms, and in my case I had to eliminate chicken eggs, dairy, tomatoes, and corn, at least for a while (I was able to add them back to my diet after being gluten-free a few years, once my villi healed).
To begin an elimination diet, it makes sense to start with the top most common food allergens, as identified by regulatory agencies like the FDA (U.S. Food and Drug Administration) and health organizations like the CDC (Centers for Disease Control and Prevention), and eliminate them one at a time for 2-3 weeks, then add the item back and record any symptoms or issues you might have. It might make sense to start this process in this order:
Milk
Eggs
Peanuts
Tree nuts (such as almonds, cashews, walnuts)
Soy
Fish
Shellfish (such as shrimp, crab, lobster)
Sesame seeds
Mustard
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