Prepare the grill for direct cooking over medium (350° to 450°F).
One at a time, place each breast, smooth side down, between two sheets of plastic wrap and pound to an even ½-inch thickness.
Season the chicken on both sides evenly with ¾ teaspoon salt and ½ teaspoon pepper. Place the chicken, smooth side down, on a work surface. Lay 2 sage leaves on top of each piece of chicken. Then wrap 2 slices of prosciutto around each breast, staggering them slightly so the meat is completely covered and the ends meet on the other side. Press the prosciutto so that it clings to the chicken. Lightly brush both sides with oil.
In a medium skillet over medium-high heat, melt 3 tablespoons of butter. Add the shallot and cook until it just turns translucent, 3 to 4 minutes, stirring occasionally. Add the wine and bring to a simmer. Add the chicken broth and lemon juice. Bring to a boil and reduce by half, about 10 minutes, stirring occasionally. The sauce should barely coat the back of a spoon. Remove the sauce from the heat and blend in the cold butter, 1 tablespoon at a time. Season with salt and pepper and set aside.
Brush the cooking grates clean. Grill the chicken, sage side down, over direct medium heat, with the lid closed as much as possible, for 3 to 4 minutes. Using a metal spatula, gently turn the chicken over, taking care not to tear the prosciutto. Grill for 3 to 4 minutes more. To check for doneness, cut into the underside of one breast. The meat should be opaque all the way to the center. Remove from the grill and let rest for 3 to 5 minutes. Warm the sauce over medium heat.
Serve the chicken warm with the sauce.