Remove chicken tenders from each breast and cover breasts with a sheet of plastic wrap. With a mallet, pound breasts flat to about 1/8". Cut each breast in half crossways. Season chicken with fresh ground pepper.
In a large skillet heat butter over medium heat. When butter is foaming, add whole sage leaves and saute on both sides for about a minute.
Place a sage leaf on each breast piece and place each breast on a slice of prosciutto. Tuck sides in and roll up to create packages wrapped in prosciutto.
Reheat skillet with butter over medium heat.
Dip chicken packages in beaten egg and then place in skillet. Cook chicken packages on both sides for about 5 to 7 minutes per side. The chicken should be done but not dried out.
Remove chicken to a baking sheet. Drape a slice of cheese over each chicken package and set oven to broil and place oven rack on the second setting from heat source. Broil until cheese melts.