Season the turkey thighs liberally on all sides with the kosher salt and pepper and place on a plate or in a shallow baking dish along with the bay leaves and thyme. Refrigerate, covered, for at least overnight and up to 2 days.
Preheat the oven to 200 degrees F.
Remove turkey from the refrigerator and rinse well under cool running water. Pat turkey dry with paper towels. Scatter the bay leaves, garlic, thyme, and rosemary along the bottom of a high-sided baking dish or Dutch oven just large enough to hold the turkey thighs in 1 layer and add the turkey thighs. Add enough oil to cover by 1-inch. Place in the oven, uncovered, and cook overnight, until meat is fall-from-the-bone tender, usually about 15 hours. Transfer to a cooling rack and allow to cool to room temperature. Remove the thighs from the oil and pull the meat from the bones. Discard bones and skin. Pull or coarsely chop meat into bite size pieces and set aside until ready to prepare pasta. Strain and reserve confit oil, rosemary sprigs, and garlic cloves separately. (Confit may be made up to 1 week in advance and refrigerated until needed. Reserved confit oil may be refrigerated for up to 1 week as well and can be used in place of olive oil for savory applications.)
In a large skillet over medium-high heat add 1/4 cup of the reserved confit oil and, when hot, add the onions and cook until softened, about 3 minutes. Add the mushrooms, 1/2 teaspoon of the kosher salt and 1/8 teaspoon of the pepper and cook, stirring, until lightly caramelized, about 4 minutes. Mash enough of the reserved confit garlic to measure 1 tablespoon and add this to the skillet along with the freshly minced garlic. Cook, stirring, for 1 minute. Add the reserved turkey confit and the al dente pasta and 1 cup of the reserved pasta cooking water, the remaining 1/2 teaspoon of kosher salt, and remaining 1/8 teaspoon black pepper and cook, stirring, until liquid is mostly evaporated and pasta and turkey are heated through, about 4 minutes. Add the parsley, stir to combine, and serve immediately, in shallow bowls, sprinkled with Parmesan, to garnish.