Prepare the chicken
Preheat the broiler (grill). Cut each chicken breast half in half lengthwise to make 8 pieces total. Place each piece of chicken between 2 sheets of plastic wrap and lightly pound with a meat mallet or the bottom of a heavy pan to an even thickness of about ¼ inch (6 mm). Season the chicken with salt and pepper, then sprinkle with the sage.
Cook the chicken
In a large frying pan over medium-high heat, melt 3 tablespoons of the butter. Add the chicken and cook until golden, 2–3 minutes. Turn and cook until just opaque throughout, about 3 minutes. Top each piece with 1 cheese slice and 1 prosciutto slice. Transfer to a baking sheet and cook under the broiler to melt the cheese, about 1 minute.
Make the sauce and serve
Add the remaining 1 tablespoon butter to the pan. Add the shallots and sauté until softened, about 1 minute. Add the Marsala and cook, stirring to scrape up the browned bits on the pan bottom, until the sauce is slightly reduced, 1–2 minutes. Squeeze in 1 teaspoon lemon juice and stir to blend. Transfer the chicken to plates, top with the shallot sauce, and serve.