1. In a 9- by 13-inch pan, mix vinegar, oil, pepper, and 1/2 teaspoon salt.
2. Cut potatoes into 1/4-inch-thick slices; add to pan and mix to coat.
3. 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.
4. 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).
5. 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.
6. Thinly slice basil leaves; add to vegetables in pan and mix gently.
7. 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.