Place chicken between 2 sheets of plastic wrap on work surface. Using mallet (or the flat end of a rolling pin), pound chicken to 1/3-inch thickness. Sprinkle chicken generously with salt and pepper. Divide sage evenly atop each chicken breast half. Top each chicken breast with 2 prosciutto slices, pressing firmly to adhere prosciutto to the chicken. On a shallow plate, put ½ cup of flour. Dredge chicken through the flour, on both sides, and shake gently to remove any excess.
Melt butter with oil in large nonstick skillet over medium-high heat. Add chicken, prosciutto side down and cook for 4 minutes. Turn chicken over and cook just until cooked through, about 3 more minutes. Transfer chicken platter and cover to keep warm. Do not rinse out the skillet. You want all the good bits from the cooked chicken to make the sauce.
In a small bowl, whisk wine together with the remaining flour. Add broth and lemon juice to same skillet; bring to boil. Add wine mixture; whisk until sauce thickens slightly, about 30 seconds.
Season to taste with salt and pepper. Spoon sauce over chicken and serve immediately.
*Sage is a controversial herb – you either love it or hate it. If you are in the latter camp, you can replace with thyme, parsley, or rosemary.