While the turkey roasts, toss the potatoes in a large pot of cold water. Add 1 tablespoon salt, cover, and bring to a boil over high heat. Cook until tender, 10 to 15 minutes. Drain the potatoes, return half of them to the pot, and mash well with a handheld masher or a fork. Set the rest of the potatoes aside and cover to keep warm.
In a medium skillet over medium-high heat, warm the butter until it melts and the foaming subsides. Crumble in the ground sausage and cook, breaking up the clumps, until well browned, 10 to 12 minutes.
Using a slotted spoon, scoop the sausage into the pot with the mashed potatoes and set that aside for the moment. Reduce the heat under the skillet to medium and, if the skillet seems dry, add a little more butter. Add the onion and cook, stirring occasionally, until golden, about 12 minutes. Stir in the garlic and cook, stirring, for 1 minute more. Add the chopped reserved liver and cook, stirring, until browned, about 3 minutes more. Scoop the mixture into the pot with the mashed potatoes.
Whisk the yolks and milk into the potato mixture until smooth; if the dressing seems too thick, whisk in more milk. Place the pot over medium heat and stir until the yolks are cooked, about 3 minutes. Fold in the reserved potatoes, sprinkle in the nutmeg and parsley, and season well with salt and pepper to taste. Keep warm.