Yes it does say that - I must confess I've fallen out of the habit, which is maybe one of the reasons I fell off the bandwagon.
Hazy memory tells me it's to do with setting your brain up to increase seratonin levels the next day, similar to what some ADs do. I'm not the best at retaining scientific information (nor spotting what might be cod-medicine), but I followed the advice anyway and it did make a difference.
You find, over time, that you eat less carbs with your evening meal because you know the spud treat is coming up. Plus it avoids the sugar drop in the morning, where you're climbing the walls to eat and snapping at everyone until after breakfast. You can have a drink of water and a shower, get the packed lunches ready first. It also helps you to last until lunchtime without snacking, because you've eaten breakfast later.
Beans on toast would be a good veggie breakfast, maybe with cheese on top.
I'm quite fond of toast with peanut butter - but you have to load it on to get enough protein. And I sometimes have muesli which I make myself and add extra chopped nuts to, or eat some brazil nuts afterwards to bulk it out.
Cottage cheese could be your friend! Veggie sausages obviously.