How to Make It
In a medium saucepan, bring the water and stock just to a simmer over moderately high heat. Keep hot over very low heat.
In a large pot, heat the olive oil. Add the sausage and cook over moderately high heat, breaking up the meat with a wooden spoon, until browned and just cooked through, about 8 minutes. Using a slotted spoon, transfer the sausage to a bowl.
Add the onion, carrot, celery and a generous pinch of salt to the pot and cook over moderate heat, stirring occasionally, until softened and just starting to brown, about 8 minutes. Add the mushrooms and a generous pinch of salt and cook, stirring occasionally, until tender, about 8 minutes.
Return the sausage to the pot, then stir in the polenta, thyme, sage and bay leaves. Very gradually stir in the hot stock and bring to a boil. Simmer over moderately low heat, stirring frequently, until the polenta is tender and thick, 45 to 50 minutes. Pick out and discard the thyme, sage and bay leaves. Stir in the cheese and season the polenta with salt and pepper.
Spoon 3 cups of the polenta into a food processor and puree until smooth. Transfer to a bowl and let cool slightly. Scrape the remaining polenta dressing into a large baking dish, cover with foil and keep at room temperature.
Preheat the oven to 425°. Set a flat rack on a large rimmed baking sheet and spread the thyme and sage sprigs on it. Run your fingers under the turkey’s breast and thigh skin to loosen it, then stuff the pureed polenta under the skin. Brush the turkey with olive oil and season with salt and pepper. Place on the rack breast side up and let stand at room temperature for 45 minutes.
Roast the turkey for about 1 hour and 40 minutes, until an instant-read thermometer inserted in the thickest part of the breast registers 155° and in the thickest part of the thigh registers 165°. Transfer to a carving board and let rest in a warm place for 30 minutes.
Meanwhile, in the oven, warm the polenta dressing in the baking dish for 20 minutes. Carve the turkey; serve with the dressing.