Step 1

Bring water to a boil in a large pot over medium-high. Stir in 1/4 cup of the salt, and return to a boil. Add potatoes; cook, stirring occasionally, until fork-tender, 12 to 15 minutes. Remove from heat. Using a spider or large slotted spoon, transfer potatoes to a colander; drain. Let cool to room temperature, about 1 hour. Reserve potato cooking water in pot.

Step 2

While potatoes are cooling, melt 2 tablespoons of the butter in a large skillet over medium-low. Add onion and 1/4 teaspoon of the salt; cook, stirring often, until onion is golden brown, 10 to 12 minutes. Set aside, and let cool 10 minutes.

Step 3

Pulse panko in a food processor until finely ground, 6 to 8 pulses; set aside. Transfer cooled potatoes to a large bowl; mash using a potato masher until smooth. Stir in cooled onion, egg, and 2 teaspoons of the salt until well combined. Stir in flour using a fork until just combined. Gently knead just until dough forms a ball. Transfer dough to a lightly floured work surface; divide evenly into 4 pieces (about 8 ounces each). Roll 1 dough piece into a 1/2-inch-thick rope (about 36 inches long), dusting with flour as needed to prevent sticking. Cut rope crosswise into 1-inch-long pieces. Transfer pieces to a lightly floured baking sheet. Repeat process with remaining 3 dough pieces.

Step 4

Return potato cooking water in pot to a boil over medium-high. Melt 5 tablespoons of the butter in a large skillet over medium. Add half of the panko (about 3/4 cup) and 1/4 teaspoon of the salt; cook, stirring often, until panko is golden brown, about 3 minutes. Remove from heat. While panko is cooking, add half of the dumplings to boiling water; cook, stirring occasionally, until dumplings are tender and float to the surface, 2 to 3 minutes. Using a slotted spoon, transfer cooked dumplings to panko in skillet; toss to coat. Season to taste with additional salt; if desired, garnish with chives. Serve immediately. Wipe skillet clean. Repeat cooking process with remaining 5 tablespoons butter, 3/4 cup panko, 1/4 teaspoon salt, and dumplings.