½ lb. (2 sticks) unsalted butter, cut into pieces
4 large russet potatoes, peeled
1 to 1½ cup flour
½ tsp. kosher salt
1 egg, beaten
• Melt the butter in a heavy-bottomed, stainless-steel skillet over medium heat, whisking frequently. The butter will foam, then subside. Continue to whisk as light brown specks form at the bottom of the pan. When the butter is browned and smells nutty, about 3 to 4 minutes, remove from heat and pour into a bowl. Set aside and keep warm.
• Place the potatoes in a large pot of salted water and bring to boil over medium-high heat. Lower the heat to medium and cook until tender, 40 to 45 minutes. Drain and let cool.
• Roughly chop and mash well with a fork or run the potatoes through a ricer. Add 1 cup flour and the salt and knead until a smooth dough forms, adding flour 1 tablespoon at a time if dough is too sticky. Mix in the egg until incorporated.
• Use your hands to form ¼ cup dough into balls a bit larger than a golf ball.
• Bring a large pot of salted water to just below a boil over medium-high heat. Working in batches, add 4 to 5 dumplings to the pot and cook 10 to 15 minutes, until the dumplings rise to the top. Using a slotted spoon, transfer to a serving platter and keep covered with a damp towel. Repeat with the remaining dumplings.
• To serve, pour the brown butter over the dumplings.