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Preheat the oven to 425ºF.
Place a baking rack on a sheet pan and arrange the sausages in one layer on the baking rack. Bake for 18 to 20 minutes, until the sausages are just cooked through.
Meanwhile, place the potatoes in a large saucepan with 1 tablespoon of salt and enough water to cover the potatoes. Bring to a boil and simmer for 20 to 25 minutes, until the potatoes are very tender. Drain the potatoes in a colander and return them to the saucepan. Add the butter, crème fraîche, milk, Dijon mustard, whole-grain mustard, dry mustard, 1 tablespoon salt, and the pepper and beat the potatoes in the pan with a handheld mixer on low speed until very smooth and creamy.
To serve, mound a generous portion of potatoes on a dinner plate and top with sausages that have been cut in half diagonally. Sprinkle with fresh chopped parsley and serve.