Position rack in the lowest position; preheat oven to 400° F.
Place oil and onion in a large skillet over medium heat. Cook, stirring occasionally and reducing heat as necessary to prevent scorching, until softened and starting to turn golden, about 20 minutes. Stir in vinegar.
Meanwhile, combine rosemary, sage, salt, pepper and nutmeg in a small dish. Place turkey in a medium bowl and sprinkle the spice mixture over it. Gently knead to combine. Form into 4 equal balls. Rub mushrooms on both sides with Worcestershire. Place one turkey ball in each mushroom cap, patting down to fill the cap. Place on a baking sheet.e the stuffed mushrooms on the bottom rack until the mushrooms begin to soften, about 20 minutes. Remove from the oven, spread the onion mixture over the filling and top with cheese. Continue baking until the cheese is melted and golden and the turkey is cooked through, about 10 minutes more.