1. Slice the vegetables no thicker than 1/4-inch (.75cm). Sprinkle the eggplant and the zucchini with salt. (See headnote whether or not you should salt the zucchini. If not, just toss the slices with a bit of olive oil and salt, then grill them as is.) Place the sliced vegetables in a mesh colander and let them drain for 30 to 60 minutes, turning them a few times to encourage the liquid to run off.
2. Wipe the vegetables dry then toss in a little bit of olive oil.
3. Brush a grill pan, or a grill, with olive oil and when hot, grill off the sliced vegetables, getting them nicely charred on one side before turning them over. Generally speaking, they’re ready to flip when they become translucent in the middle. (You don’t want crunchy or undercooked vegetables.)
4. As you grill off the vegetables, put them in a wide serving dish, and continue to grill off the rest of the vegetables. If the vegetables slices start sticking, brush more oil on the grill pan or grill as you go.
5. Once the vegetables are done, make sure they are arranged nicely in the dish, then make the vinaigrette by mixing the vinegar in a bowl with the mustard and a small sprinkle of salt; since the vegetables have been salted, go very easy on the salt here – you can add more later, to taste. Add the olive oil, za’atar, and sumac (if using) to the dressing, then pour it over the vegetables.