Pat the chicken dry with paper towels and season with salt and pepper, evening dividing them between both sides of all the cutlets. Add the flour to a medium, shallow bowl and dredge the cutlets in flour, shaking off excess before setting the cutlets aside.

Add 2 tablespoons of the butter and the oil to a large heavy skillet set over medium heat. As soon as the foam subsides, add the cutlets in a single layer (you may have to do this in batches). Place two or three sage leaves on top of each cutlet — however many fit without overlapping — and sprinkle with the Parmesan, dividing it equally among all of the cutlets. Top the sage leaves and Parm with a slice or two of prosciutto, pressing down on the prosciutto with a spatula. Cook for 3 minutes.

Carefully flip the cutlets and press down on them, helping the Parmesan cheese melt so that the prosciutto can adhere to the chicken. Cook 3 to 4 minutes before transferring the cutlets to a plate, prosciutto side up. If you have to cook your cutlets in batches, note that you may need to add an extra drizzle of oil and slightly reduce the heat to prevent burning, which may impact cooking time for the second batch.

Add the remaining 2 tablespoons of butter and the garlic to the hot skillet. Cook until fragrant, about 1 minute. Add the stock and bring it to a boil. Reduce the heat to medium-low (or whatever heat allows you to maintain a gentle simmer), and reduce the sauce by half again, 3 to 4 minutes. Add the lemon juice and return the chicken to the pan, prosciutto side up, along with any accumulated juices. Cook for 2 minutes longer, and serve immediately with sauce spooned over the top.