Make the yogurt sauce by combining the yogurt, cilantro, lime juice, garlic, hot sauce, salt and pepper in a bowl. Refrigerate until you need it.
Brush the sliced zucchini, yellow squash, eggplant, onions and mushrooms with olive oil and sprinkle with salt and pepper. Grill in batches until soft and tender, about 4 minutes per side.
Arrange the grilled veggies on a platter and serve with the sauce