양송이 is a kind of mushroom. 송이버섯 ("button mushroom") is traditionally common kind, and the prefix 양 (originally means ocean) is used for things coming from abroad, especially the West. So 양송이 is probably "western button mushroom". 양송하다 is not an existing word. The speaker is making up a new word on the fly to mean "in a manner suitable for the soft mushrooms" - something like "softly and crisply". So it means "One should slice the button mushrooms the way they deserve to be with new ...".