How are savoury meringues made?
The eggs whites in traditional meringues are used to spread the sugar into a thin foam that is then dried in the oven (or dehydrator) leaving behind the sugar structure and some proteins from the eggs. To make a meringue you need something that dissolves to tangle up with those proteins. I would guess that the beet meringues from Café Atlantico are made with beet powder replacing the sugar with the goat cheese in the middle adding to the sense of savory.

I have never made savory meringues, but if I were to experiment I would mix freshly whipped eggs whites with sugar to those created by reducing a savory liquid and then adding powder egg whites to it.