For the Chicken Cutlets: Generously season both sides of each breast with salt and pepper and dredge one at a time in flour placed in a shallow dish; shake gently to remove excess flour; set aside.
Heat butter and vegetable oil in 12-inch heavy-bottomed skillet over high heat, swirling to melt butter. When foam subsides and butter begins to color, place cutlets in skillet, skinned side up. Reduce heat to medium-high and sauté without moving until nicely browned, about 4 minutes.
Turn cutlets over and cook on other side until meat feels firm when pressed and clotted juices begin to emerge around tenderloin, 3 to 4 minutes. Leaving fat in skillet, transfer cutlets to plate, cover loosely with foil and keep warm in 200-degree oven while preparing the sauce.
For the Sauce: Set skillet over medium heat. Add garlic and sage leaves; sauté until garlic is fragrant and sage crisps, 1 to 2 minutes.
Add lemon juice and bring to boil, scraping up browned bits from bottom of skillet with wooden spoon. Add chicken stock, increase heat to medium-high and bring to boil, stirring occasionally, until reduced to 1/4 cup, about 8 minutes. Off heat, swirl in butter and add prosciutto. Season to taste with salt and pepper. Spoon sauce over cutlets; serve immediately.