You can find salt cod in the seafood department of your grocery store or at a local fish shop. It may smell briny but once soaked will mellow while staying full of flavor. This recipe is from Larsen’s Fish Market, a tiny little seafood shack in the tiny little fishing village of Menemsha, Martha’s Vineyard.
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Place salt cod in a medium pot and pour in cool tap water to cover. Bring to a boil; drain . Repeat process until a small flake of salt cod no longer tastes inedibly salty, 1 or 2 more times. Drain well. Wipe out pot and reserve.
Meanwhile, fill a large pot with cold water. Add potatoes and bring to a boil. Reduce heat and simmer until potatoes are very tender, about 15 minutes; drain well. Transfer to a large bowl and mash with a potato masher.
Pulse onion in a food processor until a paste forms. Melt butter in reserved pot over medium heat. Add onion paste and cook, stirring occasionally until water has mostly evaporated and onion is translucent, about 5 minutes; stir in ¾ tsp. pepper. Add onion mixture to potatoes and mash until evenly incorporated.
Add salt cod to food processor (no need to clean) and pulse until finely chopped. Add fish to potato mixture and mash well to combine. Taste and season with salt and more pepper if needed. Let cool, then fold in egg. Cover and chill mixture until cold, at least 1 hour and up to 12 hours.
Form mixture into ten 4-oz. patties. Place flour in a shallow bowl and dredge each patty lightly in flour, shaking off any excess.
Pour oil into a large Dutch oven or other heavy pot to come 2” up sides. Clip thermometer to side of pot and heat oil over medium until thermometer registers 350°. Working in batches, fry cod cakes, turning halfway through, until golden brown, about 5 minutes per batch. Transfer to a wire rack as you go. Season cod cakes with salt and serve with tartar sauce and lemon wedges.
Do Ahead: Cod cakes can be formed into patties 2 days ahead. Cover and chill until ready to dredge and fry.