In a large skillet, melt the butter over medium heat. Add the onions and a pinch of salt and cook, stirring occasionally, until caramelized, about 25 minutes.
Meanwhile, preheat the oven to 375 degrees F.
Heat 1 1/2 inches of oil in a large cast-iron skillet over medium heat until shimmering. Working in batches, fry the potatoes, flipping once, until just tender and beginning to brown around the edges, 3 to 5 minutes. Remove the potatoes to a baking sheet lined with paper towels to drain, then remove the paper towel. Sprinkle the potatoes with the chives, 2 tablespoons of the vinegar and salt and pepper to taste. Toss gently to combine.
Roll your pie dough out into a 13-inch circle and place it on a baking sheet lined with parchment. Whisk the egg with a splash of water in a small bowl and brush it onto the dough, leaving a 2-inch perimeter. In the center of the dough, layer up the toppings in this order: half the onions, half the Parmesan cheese and then half of the potatoes, in a fanned even layer. Top with the remaining onions, remaining cheese, the other half of the potatoes and any additional chives or black pepper from the bottom of the pan. Brush the edges of the crust with egg and then fold them up around the potatoes. Brush the outside edges with egg and sprinkle with flaky salt.
Bake until the pie crust edges are golden brown, about 25 minutes. Sprinkle the top of the galette with the remaining tablespoon of vinegar before serving.