I was going to suggest BA could offer up a Malaysia Airlines style Nasi Lemak for breakfast but then I remembered it comes with a hard boiled egg - doh! But I also suspect it a) it wouldn't appeal to BA's average customer base outside of the odd Asian route, and B), BA won't pay Do&Co enough to make a decent version of it.
I am not massively keen on eggs on planes but I've always thought the cold plate CE breakfast dishes like smoked salmon and charcuterie were pretty decent. So if you take them, other than offering a large fruit plate I'm not sure what else can be offered for breakfast time that won't have eggs in it. A large bowl of cereal with milk would no doubt create howls of outrage on both this forum and social media. Best bet is to do what I do and avoid early morning flights unless there's no alternative!