Makes 10 servings
Whisk together juices, vinegar, and oil in a large bowl. Toss mushrooms with vinaigrette and marinate 15 minutes. Transfer mushrooms to another bowl with tongs, reserving vinaigrette.
Prepare grill for cooking. If using a charcoal grill, open vents on bottom of grill, then light charcoal. Charcoal fire is hot when you can hold your hand 5 inches above rack for 1 to 2 seconds. If using a gas grill, preheat burners on high, covered, 10 minutes, then reduce heat to moderately high.
Grill mushrooms in 3 batches on oiled grill sheet set on grill rack, with grill covered only if using gas grill, turning frequently, until golden brown, about 5 minutes per batch. Transfer to vinaigrette as grilled, then toss with sea salt.
•If your sea salt is very granular and pebblelike, crush it using the flat side of a large heavy knife or the bottom of a heavy skillet. •If you're unable to grill outdoors, mushrooms can be grilled in batches in a lightly oiled well-seasoned ridged grill pan over moderately high heat, turning frequently, about 5 minutes per batch.