Pat the chicken breasts dry and cut each one in half crosswise into two smaller, more manageable pieces. Place a half between 2 pieces of plastic wrap and, using a mallet or the back of a heavy cleaver, pound it on the underside to ¼ inch thick. Repeat with the other halves. Line up the pounded chicken on a baking sheet or cutting board and season lightly with salt and pepper. (The prosciutto is salty on its own, so the chicken will not need much seasoning.)
Wrap a piece of prosciutto around each piece of chicken, the two ends meeting on the smooth top side of the chicken. (If the prosciutto slice is too short to reach all the way around, wrap a second full or half slice around the chicken.) Lay a sage leaf across the top where the prosciutto ends meet and secure with a toothpick. Be careful not to tear the sage leaf.
Place the flour in a large, shallow dish. Dredge the chicken in the flour, tapping off any excess.
In a heavy-bottomed skillet, heat the olive oil over medium-high heat. When the oil is hot but not smoking, add 2 or 3 pieces of the chicken, toothpick side down. Cook until the chicken is browned and the prosciutto is crisp, about 4 minutes. Turn over and cook for an additional 3 minutes, until just cooked though. Transfer to a warmed plate to rest. Continue with the remaining chicken, adding about 1 tablespoon oil between each batch if necessary.
When all of the chicken is cooked, pour the white wine into the skillet, followed by the chicken stock. Stir with a wooden spoon, scraping up any bits stuck to the pan. Simmer until the liquid is reduced by one-third, 5 to 6 minutes. Over very low heat, add the butter, 1 piece at a time, and swirl continuously until melted and emulsified.
Gently remove the toothpick from each piece of chicken; if necessary, use tongs or tweezers to twist and pull out the toothpicks. The sage leaf should have adhered to the prosciutto. Pour any juices that have accumulated on the plate into the sauce and swirl to combine. Place 2 pieces of chicken on each of 4 warmed plates and pour the sauce on top. Serve immediately.