Prepare grill for direct heat set on high. Cut potatoes into 1/4 slices and place in microwave safe bowl. Cover with plastic wrap poked with a few holes and microwave potatoes for about 5 minutes to soften. You don't want to cook them completely, just get the process starting, otherwise it takes forever on the grill!

Combine ingredients for vinaigrette. Pour over potatoes, and mix until potatoes are covered. Oil the inside of a grill basket and add potatoes, reserving marinade.

Grill potatoes over medium heat, turning basket often until golden brown (about 5-8 minutes) and soft when pierced with a fork. Remove potatoes to serving platter.

Add asparagus to reserved marinade, toss to coat. Place asparagus in oiled grill basket. Grill asparagus until soft when pierced with a fork, turning often.

Add asparagus to potatoes.

Lay basil leaves on top of one another and roll them up together into a bundle. Slice through in thin slices creating ribbons. (This is called a chiffonade.)

Top Potatoes and Asparagus with parmesan cheese and basil.

May be served warm or room temperature.