Cut each turkey breast in half horizontally, but don’t cut quite all the way through. Open up the breast like a book, so it lays flat. Slice the rind off the cheese and cut into four slices.
Season the meat and press two slices of cheese and half the ham on each breast. Reshape, enclosing the cheese and ham entirely.
Preheat the oven to 375ºF (190ºC). Sauté the mushrooms in the oil and butter for 5 minutes, until softened. Season and place on the base of a small roasting pan. Lay the turkey on top.
Cover loosely with parchment paper or lightly greased sheet of foil and bake for 20-25 minutes until the meat feels firm. A little cheese may have seeped out.
Remove the turkey to a warm plate and let stand. In a roasting pan sauté mushrooms in the wine until reduced by half, then stir in the cream. Season.
Cut the wrapped turkey breasts into four diagonal slices each. Transfer these to four warmed plates, spoon the mushroom sauce over and serve.
This dish is particularly good with German Spätzle, which can be made ahead then reheated in butter with chopped parsley.