Pat the chicken breasts dry with paper towels. With a very sharp knife, make each breast into a little pocket by cutting a horizontal slit into one of the thin sides. The slit should be almost but not quite the length of the chicken breast and go almost but not quite all the way through to the opposite side. Stuff 1/4 of the provolone cubes into each breast and press the chicken lightly to seal around it. Don’t worry if it’s not perfect. Oozing cheese isn’t a terrible thing. Wrap each stuffed chicken breast in a couple of slices of prosciutto, covering as much of the surface as you can.
Heat 1 tablespoon of the olive oil over medium-high in a 12-inch nonstick skillet. Add wrapped chicken breasts and cook, uncovered, on one side until prosciutto is darkened in color and chicken breast is about halfway cooked through. Flip once and cook until chicken breast is jus cooked through. Remove chicken breasts to a plate.
Heat remaining tablespoon of olive oil in the same skillet. Add shiitakes and cook, stirring occasionally, until lightly browned and tender. Add shallot and garlic and cook, stirring frequently, for a minute or two until softened.
Pour Marsala into pan and stir to scrape up any browned bits on bottom of pan. Simmer until liquid has just evaporated. Add chicken stock and sage and simmer briskly for 5 minutes. Meanwhile, in a small bowl, stir together butter and flour with a fork to form a paste. Stir paste into pan sauce and continue to simmer for 5 minutes more. Add chicken back to pan and simmer until warm. Grind some pepper over everything and taste for salt.
To serve, place a piece of chicken on each of four plates and top with some mushrooms and sauce. Sautéed spinach makes a great, easy accompaniment.