How to Make It
In a large skillet, bring the chicken stock to a boil. Add the leeks and a pinch each of salt and pepper. Cover partially and simmer over low heat, carefully turning the leeks once, until tender, about 10 minutes. Using a slotted spoon, transfer the leeks to a plate. Pour the stock into a large glass measuring cup.
In a large saucepan, cover the potatoes with water. Add a large pinch of salt and bring to a boil over moderately high heat until tender, about 12 minutes. Pour off the water. Set the saucepan over the hot burner and shake the potatoes for 1 minute to dry them out. Pass the potatoes through a ricer into a medium saucepan. Gently stir in the butter, then stir in the milk. Season the potatoes with salt and pepper and cover.
In the large skillet, heat the oil until shimmering. Add the sausages and cook over high heat until lightly browned, about 1 minute per side. Pour in the reserved leek cooking broth, cover and simmer over moderately low heat for 5 minutes. Transfer the sausages to a warmed platter and cover with foil.
Boil the broth in the skillet over high heat until reduced to 2 cups, about 3 minutes. Add the cream and boil until reduced by one-third, about 10 minutes. Stir in both mustards and season with salt and pepper. Add the leeks to the sauce and simmer over low heat until heated through.
Rewarm the potatoes, still covered, over moderately high heat, stirring once. Spoon the potatoes onto plates and set the sausages and leeks next to them. Spoon the mustard sauce over the leeks and potatoes, sprinkle with the parsley and serve.