It seems the article has explained for itself: "sharp" refers to "kind of tangy and a little spicy." But I am not sure about this.
Context:
The American Grilled Cheese Kitchen, San Francisco
Russo admires how this restaurant takes one item and makes it incredibly well. She recommends the Piglet, made with
sharp cheddar, ham, rosemary butter and apple mustard. "It's
kind of tangy and a little spicy. For something with a name like the Piglet, it's a pretty sophisticated sandwich." And the setting is appealing, too. "This is really a fun place. It's very open and airy and has an industrial chic look." 415-243-0107;theamericansf.com
More:
http://travel.usatoday.com/destinations/10great/story/2012-04-05/10-great-places-to-bite-into-a-surprising-sandwich/54065136/1