Step 1

In a saucepan, combine potatoes and enough cold water to fully submerge them. Bring to a boil. Season with 2 teaspoons kosher salt and boil until potatoes are easily pierced with the tip of a knife, 10 to 12 minutes. Drain; let potatoes cool 5 minutes, then press through a ricer or colander and stir until smooth (or mash until smooth). Transfer to a large bowl; let cool completely.

Step 2

Stir egg yolks, scallions, lemon zest and juice, flour, 2 teaspoons kosher salt, and 1/4 teaspoon pepper into mashed potatoes. Fold in salmon. Form mixture into golf-ball-size rounds, then flatten into 1/2-inch-thick disks. Transfer to a parchment-lined tray and refrigerate until cold, at least 2 hours or, loosely covered, up to 1 day.

Step 3

Whisk whole eggs in a shallow bowl and season with kosher salt and pepper. Place panko in another shallow bowl and season. Pour a scant 1/2 inch oil into a medium heavy-bottomed skillet, such as cast iron. Heat over medium until oil shimmers and bubbles form instantly around a breadcrumb dropped in (350°F on a deep-fry thermometer). Working in batches, dip croquettes in egg, allowing excess to drip back into bowl, then coat in panko, pressing to adhere.

Step 4

Fry, flipping once and adjusting heat as needed to maintain a level temperature, until crisp and golden brown, about 5 minutes. Drain on a wire rack set over a rimmed baking sheet, then sprinkle with flaky salt; keep warm in a preheated 200°F oven while frying the rest. Serve warm, with dipping sauces.