As it mentioned in the other thread, often when it shows a menu it's linking to a 3rd party site like GrubHub or Locu. Most of those you have to pay to get your menu listed and/or pay a booking fee if someone uses that site to order.
The only way I know of to get your own menu featured, so you don't have to pay anyone and so you can control everything easier, is Schema on your site. So if no website can't do it that way I'm afraid.
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