Prep the chicken. Trim any excess fat off of the chicken breasts and set them aside on a large plate.
Pound the chicken. Place a large piece of plastic wrap on a flat, dry surface and place two of the chicken breasts, smooth side up, on top. Then add another large piece of plastic wrap on top. Use the smooth side of a meat mallet to pound the chicken until it's evenly flattened, about ¼ to ½-inch thick. Repeat this with the remaining two chicken breasts. (If you don't have a meat pounder, you can use the bottom of a small sauté pan.)
Add the fillings. Turn the chicken breasts over, so the "rough" side is up, and they are lying vertically in front of you, and season the surface of each one with salt and pepper. Add about 2 tablespoons of the sun-dried tomatoes to each one, spreading it evenly. Then gently press basil leaves on top, enough to cover all but the top couple of inches in a single layer. Slice the Provolone slices in half and place them on top of the basil, again leaving the top couple of inches clear. Finally, place the prosciutto slices on top. Roll. Beginning with the end closest to you, roll the chicken as tightly, and evenly as you can, away from you. Then place each roll on its own piece of plastic wrap and wrap it very tightly, twisting the ends (like a tootsie roll). Refrigerate them for at least 1 hour.
Cook. Coat the bottom of a large non-stick skillet with olive oil and place it over high heat. Unwrap the chicken rolls and once the pan is very hot add them, seam side up. You should hear a sizzling sound when they hit the pan -- if you don't it's not hot enough yet. Use kitchen tongs to squeeze them together if you see them opening, and flip them over after just a minute or so -- this is just to brown the tops. Now with the seam side down, turn the heat to low, cover the pan, and cook until the chicken is cooked through, about 15 minutes.
Cool, slice and serve. Let them sit for about 5 minutes, then slice the rolls for a pretty presentation and serve.