Prep the chicken like a pro. Halve each chicken breast horizontally to create 8 thin cutlets. Cover with plastic wrap and pound to an even 1/4-inch thickness. Don't go crazy here, just even them out so everything cooks at the same rate.

Flour station setup. Combine flour with 1 tsp black pepper in a shallow dish. Pat the chicken dry with paper towels (this helps the flour stick), then dredge each cutlet in the flour mixture, shaking off any excess. Lay them flat and place one prosciutto slice on top of each cutlet, pressing lightly to make it stick. Secure with a toothpick if needed.

First batch cooking. Heat 2 tbsp olive oil in a 12-inch skillet over medium-high heat until shimmering. Add the whole sage leaves (if using) and cook until they just start to change color and smell incredible, about 15-20 seconds. Transfer to a paper towel-lined plate and set aside. Add 4 cutlets to the pan, prosciutto-side down, and cook until lightly browned, about 2 minutes. Flip and cook until no longer pink, another 2 minutes. Transfer to a plate and tent with foil.

Second batch time. Add remaining 2 tbsp oil to the skillet and repeat with the remaining 4 cutlets. Transfer to the plate with the first batch and tent with foil while you make the sauce.

Sauce magic begins. Soak up any excess fat from the skillet with a paper towel. Add the vermouth, scraping up all those beautiful browned bits from the bottom of the pan. Let it simmer and reduce to about 1/3 cup, which takes 5-7 minutes. Stir in the lemon juice.

Butter finish perfection. Turn heat to low and whisk in the chilled butter one piece at a time. This creates a silky emulsion that coats the back of a spoon. Off the heat, stir in the minced sage and season with salt and pepper to taste.

Final assembly. Spoon that gorgeous sauce over the chicken, top each cutlet with a crispy sage leaf if you made them, and serve immediately while everything's hot and perfect.