The occasional time I make fish cakes, or salmon cakes, or fish patties or whatever, I have always added the egg as a binder in with the ingredients when mixing.
Yesterday I came across a different way to cook them. After mixing all the ingredients you chill it for a couple of hours in the refrigerator, then after forming the patties you dip them in the egg then bread crumbs and fry about 4 minutes each side.

Doing them this way makes the outside crunchy and the inside is not soggy but soft and nice and tasty.

I served these with tinned peas [this is the north] and pureed [mashed] turnip and parsnip.

I admit to being a little sceptical about the patties not holding together without the binder in the ingredients. If you make sure the patties are well formed, once you dip them in the egg it helps keep things tight, after dipping in the brad crumbs things hold together really well, I did not have any break up while cooking