Combine quail breasts, olive oil, Italian seasoning and salt and pepper in a small bowl. Toss to coat, cover and refrigerate for 1 – 2 hours.
Lay prosciutto flat on a clean work surface or cutting board. For each slice, place a quail breast half on one end of the prosciutto. Place a piece of cheese on the quail breast and top the cheese with another quail breast half. Roll up the prosciutto snugly and secure with a skewer, making sure that you spear through both ends of the prosciutto and the cheese.
Place on a medium hot grill until lightly browned on all sides and quail breasts are just-cooked and still slightly pink.