Wash 1 1/2 pounds of mushrooms or eggplant or both. Cut off tough stems from mushrooms. Cut eggplant in half crosswise and cut each piece in half lengthwise.
If using eggplant, arrange a rack over 1 inch of water in a 6- to 8-quart pot. Lay eggplant on rack, boil over high heat, cover, and steam eggplant until soft, about 5 minutes. Let rest until cool enough to handle.
Skewer two or three mushrooms (horizontally through cap) or one piece eggplant onto each soaked 12-inch wood kabob sticks. Coat vegetables evenly with sesame or vegetable oil.
Arrange kabobs on a cooking spray coated grill rack, shut lid, and grill 6 to 8 minutes, flipping once.
In the meantime, using a food processor or blender, pulse all sauce ingredients until smooth. Serve sauce on side.