Combine all the ingredients with a whisk in a bowl. You can season with salt and pepper.
Preheat the grill, setting the burners on high. Oil the grate.
In a bowl, combine the zucchini, eggplant, and bell peppers with the oil and garlic. Season with salt and pepper. Grill the vegetables until tender. Set aside on a plate.
In a saucepan of salted simmering water, add the white vinegar. Break the eggs, one at a time, into a saucer and slip into the water. Poach for about 3 min or until the desired doneness. Normally, the yolk should still be runny. Drain the poached eggs on a plate lined with paper towels.
Divide the lettuce among four plates. Season with pepper. Add the vegetables, eggs, and cheese shavings.