Rinse 1 1/2 pounds fresh shiitake mushrooms (2 to 3 in. wide) or Asian eggplant, or a combination of both. Remove tough stems from shiitakes. Slice eggplant in half crosswise; cut each piece in half lengthwise.
If using eggplant, place a rack over about 1 inch water in a 6- to 8-quart pan. Pile eggplant on rack, bring water to a boil over high heat, cover, and steam eggplant until soft, about 5 minutes. Let stand until cool enough to handle.
Thread two or three mushrooms (horizontally through cap) or one piece eggplant onto each soaked 12-inch wood skewer. Brush vegetables lightly with Asian sesame oil or vegetable oil.
Place on a lightly oiled grill over a solid bed of medium-hot coals or medium heat on a gas grill (you can hold your hand at grill level only 3 to 4 seconds); close lid on gas grill. Cook, turning once, until tender when pierced, 6 to 8 minutes. Serve with ginger or ponzu dipping sauce.
Ginger dipping sauce: In a blender or food processor, whirl 1/4 cup coarsely chopped white onion, 1/2 cup peeled, coarsely chopped ginger, and 1/3 cup peeled, coarsely chopped sweet apple (such as Red Delicious), scraping sides of bowl as necessary, until smooth. Add 3/4 cup soy sauce, 6 tablespoons rice vinegar, 1 1/2 tablespoons Asian sesame oil, and 1/4 cup sugar. Whirl until smooth. Serve immediately or cover and chill up to 1 week. Whisk before serving. Makes about 2 cups.
Ponzu dipping sauce: In a 2- to 3-quart pan over high heat, bring to a boil 1/2 cup sake or dry white wine, 1/2 cup mirin (sweet rice wine; available in supermarkets), 1/2 cup soy sauce, 1/2 cup rice vinegar, and 1/4 cup dried bonito flakes (optional; see notes); if omitting bonito flakes, combine all other ingredients in a bowl and chill. Lower heat and simmer 15 minutes. Strain and chill until cold, about 1 hour. Stir in 3 tablespoons lemon juice. Serve immediately or cover and chill up to 1 week. Makes 2 cups.
Nutritional Analysis per skewer of shiitakes.