Step 1

Place chicken pieces one at a time between sheets of plastic wrap and pound with the flat side of a meat mallet or a rolling pin until chicken is an even 1/2 inch thick, concentrating on the thickest part of each cutlet.

Step 2

Place 1 slice of prosciutto on a cutting board, narrow edge facing you. Put a chicken cutlet smooth side-down in the middle of the prosciutto slice. Bring the ends of the prosciutto up and over the chicken to meet in the middle of the meat. Press gently to adhere. Repeat with remaining chicken and prosciutto, using the extra 2 slices of prosciutto to patch any holes or places where the edges don’t quite meet. (The chicken breasts can be prepared up to this point and refrigerated, covered with plastic wrap, for up to 2 days.)

Step 3

Combine flour, salt, and pepper on a plate. Dredge the prosciutto-wrapped chicken cutlets in the flour until evenly coated. Shake off excess flour and discard remaining flour.

Step 4

In a large frying pan, heat olive oil over medium-high heat until shimmering. Add sage leaves and cook, stirring constantly, until they’re crisp and darkened, 1 minute. With a slotted spoon, transfer leaves to a small bowl.

Step 5

Add chicken to the pan with the prosciutto seam-side up and reduce heat to medium-low. Cook until golden brown on the underside, about 5 minutes. Turn chicken over with a wide spatula and place the slices of cheese on top. Continue to cook until an instant-read thermometer inserted in meat register 165°F), 4-8 more minutes. Transfer chicken to a serving plate that will hold some sauce, cover loosely with foil, and set aside; the cheese will continue to melt as the chicken rests.

Step 6

Add vermouth to the pan and increase heat to high. Boil, scraping up browned bits on the bottom of the pan, until the liquid has almost evaporated, 2-3 minutes. Add chicken broth and continue to boil until the mixture is reduced by half, 3-5 minutes.

Step 7

Remove pan from heat and whisk in lemon juice and the butter. Season with salt and pepper to taste. Pour sauce over chicken. Scatter fried sage on top and serve immediately.