Preheat oven to 350 degrees. Combine flour and 1 tsp salt and 1 T pepper in a shallow bowl. Pat chicken breasts dry and dredge in the flour mixture, shaking off any excess. Lay the breasts flat and top with 2 sage leaves. Wrap 1 slice of prosciutto around each breast, pressing slightly to adhere.
Set a wire rack on a rimmed baking sheet to hold the breasts. Heat 2 T olive oil in a large skillet over medium heat. Add two of the chicken breasts to the heated pan, top side down, and cook until golden brown, about 3 minutes. Turn the breasts over, being careful not to dislodge the sage and prosciutto, and cook the second side until golden brown as well, about 3 more minutes. Place the cooked breasts on the wire rack, tent with foil to keep warm, and cook the remaining two breasts in the same manner. Place the breasts in the oven for approximately 25 minutes, or until the chicken is cooked thoroughly and no longer pink on the inside.
While chicken is cooking, prepare the sauce. Over medium heat, stir the vermouth into the pan, scraping up the browned bits, and simmer until reduced to about 1/2 cup, 5-7 minutes. Stir in the fresh lemon juice. Reduce heat to low and whisk in the butter, one slice at a time. Turn off heat and stir in the fresh parsley. Season with salt and pepper to taste. Check for tartness as you may want to add more lemon juice. Spoon sauce over the cooked chicken breasts. This chicken is also very nice served over a bed of long grain and wild rice pilaf. Serves 4-6.