Rinse turkey and pat dry. Place breast half, skinned side down, between sheets of plastic wrap; with a flat mallet, gently pound to an even 3/8 inch thick. Peel off top sheet of plastic wrap.
Sprinkle fontina and parmesan cheeses 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 1 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 according to instructions below.
If using charcoal briquets, mound and ignite 60 briquets on the firegrate of a barbecue with a lid. When briquets are dotted with gray ash, in 15 to 20 minutes, push equal amounts to opposite sides of firegrate. Add 5 more briquets to each mound of coals now and after 30 minutes, if grilling takes longer than that.
If using a gas barbecue, turn all the burners to high, close lid, and heat for 10 minutes. Then adjust burners for indirect cooking (heat on sides of grill not down center under food) and keep on high.
Set a drip pan on firegrate between coals or burners. Set barbecue grill in place. Set roast in center of grill, not over heat. Cover barbecue; open vents for charcoal.
Cook roast until a thermometer inserted in center of meat reaches 160°, 20 to 25 minutes.
Transfer roast to a platter and, keeping it warm, let rest 5 to 10 minutes. Garnish with sage sprigs if desired. Cut roast crosswise into 1/2-inch-thick slices to serve. Add salt to taste.
Garlic oil. In a small, microwave-safe bowl, mix 2 tablespoons olive oil, 1 tablespoon minced garlic, and 1/4 teaspoon coarse-ground pepper. Heat in a microwave oven at full power (100%) until garlic is fragrant, 45 to 60 seconds.