If you have food allergies, should you worry about cooking oils (and fats) in your food?
In general, commercial peanut oil, soybean oil, sunflower oil, palm oil are unlikely to cause reactions as they are highly refined and their protein content is nonexistent. But there can always be exceptions. It all depends on your sensitivity level. If you are extremely sensitive to even minute amounts or if your reactions are severe it would be safe to err on the side of caution.
Oils labeled cold pressed, expeller pressed, or unrefined, are more problematic from an allergy viewpoint as they will contain small amounts of protein. These are more likely to be found in “natural”, “organic”, and higher-end foods as they are generally considered healthier due to less processing and thus less loss of nutrition.
Coconut, sesame, perilla oils are generally not refined.
And for milk allergies, ghee and clarified butter’s milk protein content is much lower than butter. Again depending on sensitivity level you may tolerate none, one, or possibly even both.