I suspect the main claim is one of trade dress, which is a subspecies of trademark. As Ed suggests, this would largely be a claim that the "look and feel" of the second restaurant copy the distinctive elements of the first restaurant. While trade dress is eligible for federal registration, I seriously doubt that the first restaurant in question here sought or obtained registration. Most of the major trade dress cases in this area have dealt with chain restaurants.
As Mike suggests, patent does not appear to be implicated here. I suppose copyright could apply if certain aspects of the menu (the actual descriptions of dishes) were copied, but there are only so many ways to factually describe what's in a dish, so any copyright protection of menu descriptions would presumably be quite thin if there is even a valid copyright interest at all. While recipes themselves can be copyrighted (again, the more descriptive elements and not simply the ingredients lists), assembling a dish based on another's (copyrighted) recipe is not an infringement of that copyright. Trade secret is another potential ground (claiming that some process, procedure or formula is a trade secret), but the requirements for protecting a trade secret and for what qualifies as a trade secret set a very high bar that perhaps only the Motos of the world might reasonably be able to meet.
As the Columbia professor in the article suggests, the best claim, if any, may be based in contract. If the individual had signed an NDA or noncompete, the original restaurateur would presumably have a better case. On the whole, maybe this was against the ethics and norms of the restaurant industry, but unless there are material facts left out of the article that pop up in the complaint (such as the existence of an NDA, noncompete or some other sort of contract), it seems questionable whether there is anything legally actionable here.