To cook the saltimbocca, preheat a stainless-steel pan over medium-high heat. Once hot, add the oil, followed by the chicken, prosciutto-side down. Let cook. Shake the pan periodically and once the chicken releases on its own, you’ll know it’s time to flip it. Don’t worry if the cheese oozes out of the sides during cooking. This will make the sauce even tastier. Check for doneness and place onto a plate once done. Loosely tent with foil.
Make sure the surface of the pan is shiny with the oil that remains, adding a bit more if needed, to cook the second batch. Discard any excess oil. Off the heat, add the wine and garlic. Let simmer and reduce this to about 2 tablespoons, scraping up any browned bits from the bottom of the pan.
The sauce should look a little syrupy and have a nice sheen to it, before adding the stock. Bring this to a boil and reduce again by half. Once it looks syrupy again, turn off the heat and add the cold butter. Gently swirl until everything is melted and combined. Add the lemon juice and taste the sauce for seasoning.
Reduce the heat to low and return the chicken to the pan, pouring in any accumulated juices. Coat both sides of the chicken with the sauce. Place onto a platter, pour the sauce over top, and serve.