Preheat the oven to 400 degrees F. Scrub the potatoes, place them on a baking sheet and bake until the thickest part is easily pierced with a knife, about 1 hour 15 minutes. Remove the pan from the oven and allow the potatoes to cool completely. Break them up by hand into rough 1- or 2-inch pieces. (This step can be done a day in advance; refrigerate the potatoes until needed.)
In a small bowl, whisk to combine the mustard and vinegar. While whisking, add the olive oil in a steady stream to form an emulsion.
In a deep-fryer or pot over medium-high heat, heat about 4 inches of peanut oil or lard to 360 degrees F. Fry the potatoes in batches until golden brown and crisp, using a slotted spoon or frying spider to turn them often, about 4 minutes. When the potatoes are done, use a slotted spoon to transfer them to a paper towel–lined plate.
In a large bowl, combine the potatoes, scallions and enough vinaigrette to lightly coat (about 2 tablespoons) and toss to combine. Pour onto a platter, sprinkle with flaky salt and serve.