My favorite summer breakfast is warm pita bread smeared with labneh (yogurt cheese), drizzled with good olive oil, heavily sprinkled with za'atar and topped with perfect slices of tomato.
Cut stale pita into triangles, and pull the two halves apart. Toss the triangles in a bowl with good olive oil and sprinkle with za'atar. Add a little salt, if needed. (the za'atar I buy at a local Middle Eastern market already has salt in it. Toast in a 350 degree oven until the chips are crisp.
Make a salad of chopped cucumber, tomatoes, parsley, diced red and green sweet peppers, chopped red onion, and minced garlic. Toss with za'atar to taste. You can keep the dressing simple--lemon juice, olive oil, salt. If you throw in torn up, bite-sized pieces of toasted pita bread, it is called fattoush. Or you can get fancier and make a dressing with tahini, lemon juice, thinned with water (the tahini will "seize" at first, then smooth out as you add more liquid) and salted to taste.
Sprinkle it on soft scrambled eggs or on sandwiches of hard boiled egg and tomato slices--with olive oil, of course.
A common snack found at Middle Eastern bakeries goes by various names but it is basically a yeasted, soft flat bread dough (I think every version I've ever tasted probably had olive oil in the dough, itself) smeared with olive oil and heavily topped with za'atar, then baked.
Marinate chicken breasts or thighs in lemon juice, olive oil, garlic and za'atar. You can either broil on skewers or bake.
One of my favorite Yotam Ottolenghi recipes using za'atar is this one:
Ottolenghi uses za'atar a lot.
That's what I can think of off the top of my head.
I buy the stuff in pretty big bags from my local Middle Eastern market and always seem to be replacing it in my pantry because I have run out.