I have seen restaurants with dishes containing "Leek ashes". What do the ashes add to the dish?
I've seen references to leek ashes providing a smoky note to sauces and that this is a Spanish origin technique. Using it as a coating for steak also appears to be trendy, so I'd guess it's again for a subtle smoky flavour.
I'm most familiar with ashes being part of cheese. For example, Morbier has a layer of vegetable ash in the middle, and some goat cheeses are rolled in ash. I assume that originally this would have been for a preservative effect, but is now part of the character of the cheeses.