Place the chicken breasts between two sheets of parchment and pound until thin and even.
Season both sides lightly with salt and black pepper.
Place two sage leaves on each chicken cutlet and cover with a slice of prosciutto, pressing gently so it adheres.
Lightly dredge the chicken in flour, shaking off excess.
Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat.
Place the chicken prosciutto-side down in the skillet and cook for 3–4 minutes until golden and crisp.
Flip and cook the other side for 2–3 minutes until cooked through.
Remove chicken from the skillet and keep warm.
Add white wine to the pan, scraping up any browned bits, and let reduce by half.
Stir in chicken broth and remaining butter, simmering until slightly thickened.
Return chicken to the skillet briefly to coat with sauce.
Serve immediately with sauce spooned over the top.