Have you looked at Pepperidge Farms breads? (no english muffins with them)
That is the brand we have used for many years - including before my DD could eat baked egg (she can not eat direct egg, but has grown up to be able to tolerate egg mixed with wheat and cooked at a high temp). We have the added issue of sesame.
I trust Pepperidage Farms labels, but you should call to determine if you trust them for your child.
Major manufacturers have different bakeries in different areas of the country, it is possible for something to be safe out of one bakery that distributes to one area of the country, and not safe in another. (this has even historically happened with things like Girl Scout Cookies!)
Watch rolls as they can be brushed with an egg wash (think shiny dinner rolls). They do that for "beauty" as well as to hold seeds on the rolls.