Remove brown gills from undersides of mushroom caps using a spoon; discard gills.
Brush caps with 1 tablespoon Rosemary Oil.
Snap off tough ends of asparagus; brush with remaining 1 tablespoon Rosemary Oil.
Grill mushrooms, covered with grill lid, over high heat (400° to 500°) 5 minutes on each side.
Cut mushrooms into strips.
Grill asparagus, covered with grill lid, over high heat 2 minutes on each side.
Serve together.