Rinse turkey and pat dry. Place breast half, skinned side down between sheets of plastic wrap; with flat mallet, gently pound to an even 3/8-inch thick. Peel off the top sheet of plastic wrap.
Sprinkle fontina and parmesan cheese evenly over pounded turkey breast to within 1 1/2 inches of edges. Scatter sage evenly over cheese. Fold narrow ends of turkey over cheese mixture, then from one long side, gently roll turkey, peeling the bottom sheet of plastic wrap away as you go, to enclose filling and form a log about 3 inches wide and 12 inches long. Brush all over with garlic oil.
Wrap prosciutto slices crosswise around turkey, overlapping to cover log completely, including ends. Tie roast with cotton string at 2-inch intervals.
Prepare barbecue for indirect heat. Cook roast until thermometer inserted in center of meat reaches 160°, 20 to 25 minutes.
Transfer roast to platter and, keeping it warm, let rest 5 to 10 minutes. Garnish with sage sprigs. Cut roast crosswise into 1/2-inch thick slices to serve. Add salt to taste.