In a food processor, combine all glaze ingredients except the oil and pepper. Process until smooth. With the motor running, gradually add the oil through the hole in the lid; the glaze will thicken. Season with pepper and pulse to blend. Adjust the seasoning and pulse again just before using. Use immediately or cover tightly and refrigerate for up to 3 days. Return to room temperature and whisk to re-blend.
PREPARE THE KABOBS:
Cut each zucchini crosswise into 9 equally thick slices. Divide the vegetables, alternating them, among 6 metal or wood skewers. Be certain to arrange the flat sides of the zucchini slices and the stems ends of the mushrooms so they will get the maximum exposure to the heat.
GRILLING THE KABOBS:
Preheat a gas grill to medium-high. When the grill is ready, lightly brush the kebabs with some of the glaze. Lay the kebabs on the rack, cover, and grill, turning every 2 minutes and basting often with the glaze, until it is used up and the vegetables are lightly colored by the grill, 6 to 8 minutes total. Season with salt and pepper and slide the vegetables off the skewers onto plates. Serve hot or warm.