Sprinkle chicken evenly with sage and season lightly with salt and pepper. Put flour in a shallow dish; dredge chicken in flour and shake off excess.
In a large saute pan over medium heat, warm olive oil. Working in batches, cook chicken until golden brown underneath, about 5 min. Turn chicken over and cook until golden brown underneath, about 5 min.; transfer to a platter.
Whisk stock into pan. Add prosciutto and vinegar and bring to a boil. Reduce heat to low, return chicken to pan and cook, basting occasionally with sauce, until juices run clear when chicken is pierced with a knife, 7-10 min.
Transfer chicken to a warmed platter. Season sauce with salt and pepper and serve alongside.
