Back in the 19th century though, the brine was boiling pretty much 'round the clock. Local legend has it that salt potatoes came about when workers started cooking potatoes in the brine that was brought up from underground. Given that a large percentage of the immigrants in the area were of Irish origin, this makes particularly good sense. The high salinity in the brine yielded a boiled potato far saltier and of more interesting flavor than others—the salt at that high a level penetrates all the way into to the center, so this isn't just a standard lightly salted boiled potato, and it has a texture that's a bit more akin to a baked potato, I think.
I'm a novice salt potato maker, not an expert, so take everything I say here with, sorry, a large grain of you know what. I heard about them first from Kim Huffman, a long-time customer here who comes to town regularly but lives up in that area. She wrote me this spring: "The potatoes are served all over around here. I can't tell you how many roadside chicken bbq's I've been to that have them." "Another interesting aspect of cooking them," she added, "is that way is that by saturating the potato it stays whole and firm and doesn't become flinty or crumbly as typical." To convey the full salt potato experience, she added, "Of course a ladle of melted butter over 3 or 4 (or more!) golf ball-sized potatoes is essential."
I'm sure some upstate New York natives will be able to add to my mix, but basically you cook new potatoes—which are out in the market here en masse this month—in very salty water. I don't mean water where you add a bit more salt than you usually do. I mean like where you have about two quarts of water dosed up with over a pound of salt to cook a couple of pounds of new potatoes. There are various recipes with varied salt to spuds to water ratios out there but they all seem to be roughly at that level. I'm not the science super hero, but from what I know basically, as Kim said above, one wants to oversaturate the water with salt. Which means that you'll be adding so much of it to the water that the salt won't actually all dissolve. (I did see some recipes for "Low-Salt Salt Potatoes" but . . . seriously. . . .) One of the keys of the cooking seems to be to let the potatoes rest on a dry cloth for a few minutes after you take them, already cooked 'til they're tender, from the hot salt water. While the spuds are setting up, a sort of "frosting" of salt will solidify on the surface of the skin, looking a bit like frosted (sugared) grapes if you happen to ever have seen those. Once they've rested for a minute or two you serve them with a dish of melted butter for dunking, and eat.
As for the history, here's an 1899 newspaper report that kvells lovingly about this local specialty. You'll see that these are no recent trendy addition to upstate New York eating. Going back to written records from the Syracuse Sunday Herald from over a century ago (Sunday, 5 March 1899. Page 26 to be exact):
Of all the Bohemian resorts in the city there is but one that is distinctly a product of Syracuse. For sixteen years it has flourished in Wolf Street, and the Brothers Keeffe, who are its proprietors, are almost as well known throughout the country as was "Nick" Enzel of New York and his famous beefsteak.
The salt potatoes of Syracuse rank with the baked beans of Boston, the terrapin of Baltimore, the scrapple of Philadelphia and the frankfurters of Milwaukee. They were born with the salt industry of the city. In the immense iron kettles in which the brine was slowly boiled into salt, the urchins of the First ward prepared feasts for a king. A few potatoes, a few ears of corn, access to a salt kettle and a little patience were the only requirements.
The corn and potatoes were dropped into the kettle, a wait of half an hour would bring a rich reward to the youthful epicures. Potatoes, mealy and bursting through their skins, with the salt clinging to them in particles that shone like miniature diamonds, tickled the palate in a meal that was not to be despised.
The Keeffe brothers in their younger days boiled potatoes in the old salt kettles. When they were older and added a barroom to the queer little old grocery shop of which they were proprietors, they bethought themselves of the salt potatoes of their boyhood, and salt potatoes formed the entire bill of fare of the lunch which they served with the foaming beer.
Salt potatoes were popular with the public, and a Syracusan who entertains a stranger without giving him a chance at the delicious delicacy is lacking in some of the fine points of hospitality.
The saloon where they are served isn't much in the way of a luxurious drinking place. A plain little bar takes up one side of the small room, whose sawdust floor leads through a door into a room beyond, where a table placed among the kegs and barrels of liquers [Ed. sic] makes up the furnishings of the Cafe Keeffe.
Today it's apparently quite common to buy bags of "salt potatoes" where they pack the right amount of salt into packets stuck into each bag of potatoes to make it easier for the home cook. I've heard tell that they're typically sold at country fairs with little wooden folks to spear them with. They're typically served with a lot of melted butter, into which the potatoes are dipped, which, to my mind, makes them a very poor inland man's version of lobster, or maybe a sort of upstate New York version of Raclette. Better butter is of course a good way to go—I'd use the Kerrygold Irish cultured butter (that's the one in the silver foil) in deference to the contribution of the Irish salt workers. And because it's very, very good.