Since it opened on a high-traffic strip of Avenue A in the spring, Foxface Natural has kept a low profile, as discreet and self-contained as a sealed envelope. At times it almost seems to be hiding. Slats on the window make it hard to see inside from the street. The exterior is painted battleship gray. There is no sign.

Nevertheless, the restaurant has attracted a small and fervent following. I don’t know many people who’ve eaten there so far, at least compared with Libertine or the new I Sodi, but the ones I do know say it’s their favorite new restaurant. It has become one of mine, too.

Whether it will be yours may depend on your tolerance for eccentricity. The soundtrack is heavy on recent Australian punk bands like Amyl and the Sniffers, the Chats and Alien Nosejob, played loud. The menu demands an adventurous spirit. If you get the whole fluke, roasted in a wood oven and served in a lake of olive oil with fistfuls of chopped garlic and parsley, the fish will not be filleted for you on a tableside gueridon. You will pick the flesh from the bones yourself.

For the right diners, this will be an exciting chance to learn about flatfish anatomy. For the wrong ones, it will be a nightmare. As a dinner companion in the first category put it while he was contemplating how best to eat the dorsal fin, which had browned in the oil into a long, crunchy and, as it turned out, delicious ridge, “This is not a restaurant for Karens.”

Foxface Natural describes itself in its Instagram bio as “a natural continuation of Foxface sandwiches.” This is like calling day a natural continuation of night. It’s accurate, but it leaves a lot unsaid.

The last time I’d seen the name Foxface, it was on a sign outside what had once been the ticket office of Theatre 80 St Marks. Behind a window, sandwiches filled with things like elk osso buco or stewed tripe with ’nduja were being sold by a couple who live next door to the theater, Sivan Lahat and Ori Kushnir. They had no room for seats, but if you wanted to eat your sandwich right away they’d invite you to take it inside a tavern next door that specialized in very old film soundtracks and absinthe. This kept up for about three years before Foxface closed last September.

I haven’t seen any elk or tripe yet at Foxface Natural. But that doesn’t mean that Ms. Lahat and Mr. Kushnir have settled into a routine of chicken tenders. Roasted antelope chops appeared one night recently, and you can usually find a very good hand-chopped tartare of kangaroo. Raw kangaroo has a deeper and sometimes gamier flavor than beef, and David Santos, the executive chef, takes advantage of that by incorporating a complex spice blend inspired by Ethiopia into the purplish meat and serving it with a smoky, black purée of charred eggplant.

The meat that made the deepest impression on me was smoked Boer goat. A breed developed in South Africa, Boers are raised for eating, as opposed to milking or making into angora sweaters. Mr. Santos cooks it overnight at a low temperature in a smoker inherited from the last tenant, Harry & Ida’s Meat and Supply Co.

The pastrami that Harry & Ida’s made in it became an obsession for lovers of smoked meat. Foxface’s goat is a worthy successor to the pastrami — maybe even a natural continuation. What it most reminds me of, though, is Texas Hill Country barbecue, if pitmasters in Texas served their barbecue with things like saffron-scented tomato sauce and creamy roasted new-crop potatoes.

The mammals are likely to be the first thing most people notice about the menu. After four meals at Foxface Natural, though, I’ve come to think of it as a seafood restaurant that occasionally dabbles in antelope. Although most of the fish Mr. Santos serves are the opposite of exotic — he favors black sea bass, striped bass and a few other fixtures of the local waters— he is particular about whom he buys them from. He prefers fish that are caught on the end of a line by crews that practice the Japanese slaughtering technique known as ikejime.

This method, thought to be more humane, also keeps the fish firm. You may notice the texture in uncooked dishes like the Boston mackerel that he cured in salt and apple vinegar one night. This was a take on a sushi-bar standard, served with a very un-Japanese sweet-pepper relish spooned over a fermented-pepper purée on a plate dotted with beehive-shaped domes of ricotta. (Like the heap of salted butter that accompanies each order of sourdough bread, the ricotta is made at the restaurant.)

I am sure, too, that ikejime was one thing that made it so pleasurable to dismantle and compare the flavors and textures of different parts of the roasted fluke, or to pull the flesh from the head, throat and belly of a butterflied black sea bass. But the thing that will really win over people who appreciate fresh fish is that Mr. Santos never seems to overcook it. He’ll pull it from the fire while there’s still some pink at the bone and sticky jellies still cling to the cartilage.

Dave Pasternack used to cook fish that way when Esca was still open. Eric Ripert and a handful of other chefs still do, but many others won’t. No doubt they get tired of having customers send fish back if it’s not completely firm and almost dry. So far, Mr. Santos doesn’t seem worried about that.

Of course, diners who won’t eat slightly pink sea bass may be reluctant to sign on for certain other items at Foxface Natural, too. Boiled gooseneck barnacles, for instance, which have been compared to dinosaur toes. Or the squares of grilled pig ear topped with white strips that look suspiciously like offal. These turn out to be the pickled skirt trimmed from a scallop — shellfish offal. They are very good dragged through a salsa verde tasting of parsley and cilantro, or a second sauce, a garlic-infused emulsion based on Basque pil-pil.

Mr. Santos, who was the chef and owner of a Portuguese restaurant in the Village called Louro until it closed eight years ago, brings a wide frame of reference to his kitchen. Some of his cooking, like the pil-pil and the fluke in olive oil, is flat-out Iberian. For striped bass steaks roasted on the spine he chose a spiced tomato sauce, chraime, originally from North Africa. The black sea bass, painted with turmeric sauce and splashed with what was essentially dill pistou, seemed to be an inspired reworking of the signature dish of Hanoi’s most famous fish restaurant, Cha Ca La Vong.

Most of the main courses at Foxface Natural will feed two. When an especially large fish has come in, it may be enough for three or four. “We’re not small-plates people,” Ms. Lahat said across the counter one night.

They are, however, natural-wine people. The wines themselves are quite approachable, although the list is sometimes less than self-explanatory. The entire entry for one white poured by the glass reads “GraWü, Chourmo.” Does GraWü go down well with raw kangaroo? There is, it’s safe to say, only one place in New York to find out.

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