Notes: For flavor, Alison Cappuccio prefers to use imported Italian Parmigiano reggiano cheese; shave the cheese with a vegetable peeler.
In a 9- by 13-inch pan, mix vinegar, oil, pepper, and 1/2 teaspoon salt.
Cut potatoes into 1/4-inch-thick slices; add to pan and mix to coat.
Spritz the inside of an 11- or 12-inch square hinged grill basket with cooking oil spray (or lightly brush with oil). Lift potatoes from pan, reserving marinade; arrange in a single layer in basket and close.
Snap off and discard tough asparagus ends; add asparagus to marinade in pan and mix to coat. Spritz the inside of another 11- or 12-inch square hinged grill basket with cooking spray (or lightly brush with oil). Lift asparagus from marinade, arrange in a single layer in basket, and close (if you have only 1 basket, cook vegetables in sequence).
Prepare barbecue for direct heat (see below). When it is hot (you can hold your hand at grill level only 2 to 3 seconds), set basket with potatoes on grill; turn often until potatoes are golden brown and tender when pierced, 6 to 8 minutes total. After potatoes have cooked 5 minutes, set basket with asparagus on grill; turn often until tender when pierced, about 5 minutes. Vegetables overlap somewhat when you turn the basket; shake it to spread them out so they cook evenly. Remove cooked vegetables from grill. Dump asparagus onto a board and cut into 2-inch pieces; return to marinade. Add potatoes to marinade.
Thinly slice basil leaves; add to vegetables in pan and mix gently.
Pour mixture onto a platter. Garnish with basil sprigs. Scatter about half the parmesan over vegetables. Serve warm or at room temperature; add remaining cheese and salt to taste.
Notes: To cook vegetables concurrently, you need 2 hinged grill baskets or a cooking grate with small holes. Hinged grill baskets are sold at barbecue supply stores; choose one with handles that will fit in your barbecue. You can also use a cooking grate (sold at barbecue supply stores as well) with small holes.
To prepare grill for direct heat:
If using charcoal briquets, mound and ignite enough briquets to cover the firegrate in a single, solid layer (you'll need about 75). When the briquets are dotted with gray ash, after 15 to 20 minutes, spread them over the firegrate in a single layer. Let them burn down to the desired heat.
If using a gas barbecue, turn all the burners to high and close the lid for 10 minutes. Adjust the burners to the desired heat.
This was okay. Pain in the butt to cook on the grill even with the pototoes sliced to the right thickness. There were four of us - 2 liked it, 2 didn't.
This one is definitely a matter of preference - either you like or you don't