In a shallow bowl, stir together flour, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Set aside.
Lay 1 sage leaf lengthwise on each cutlet, then wrap a prosciutto slice around middle of each cutlet, encasing sage. Flatten with the palm of your hand to help prosciutto adhere to the chicken. Dredge cutlets in seasoned flour; tap off excess.
In a large nonstick skillet, heat 2 teaspoons oil over medium-high heat. Cook 2 cutlets until golden brown and cooked through, 3 to 4 minutes per side. Remove cutlets, and keep warm. Repeat with remaining 2 teaspoons oil and 2 cutlets.
Add wine and broth to skillet; cook over high heat until reduced by three-quarters, about 2 minutes. Remove from heat; let cool 1 minute. Add butter and minced sage; stir until butter is melted, about 30 seconds. Spoon sauce onto plates; top with cutlets. Serve immediately.