Put the flour on a large plate. Top each cutlet with a sage leaf and top with prosciutto and wrap it around the back of the cutlet, pressing gently to adhere.
Press both sides of a prepared cutlet into flour. Shake off excess, making sure sage and prosciutto stay in place. Place on a plate; repeat with remaining cutlets.
In a large skillet, heat 2 Tbsp of the oil over medium heat. Add 4 cutlets, sage side down, and cook until golden brown, 2 to 3 minutes. Flip cutlets and cook until golden brown and just cooked through, 2 to 3 minutes more. Transfer cutlets to a platter and tent with foil to keep warm. Repeat with remaining oil and cutlets. (If the browned bits on the bottom of the pan start to get really dark, lower the heat. Otherwise, the sauce will be bitter.).
Pour off any excess oil, add wine, and briskly simmer, scraping up any bits from the bottom of the pan, until wine is almost evaporated. Add broth and briskly simmer until reduced by about two thirds, about 3 minutes. Stir in butter and lemon juice; season with salt, if needed, and pepper. Pour sauce over cutlets (on a bed of arugula, if you like) and serve.