Whisk za’atar, garlic, and oil in a small bowl. Generously brush eggplant, onions and pita with za’atar oil. Season with salt and pepper; reserve remaining oil for dressing.
Prepare a grill for high heat; thoroughly clean grates and brush with oil.
Grill eggplant, turning occasionally, until lightly charred on all sides but just tender, about 3 minutes per side. Grill onion and pita, tuning once, until lightly charred on both sides, 1 to 2 minutes per side. Transfer to a rimmed baking sheet and let cool. Cut large eggplant into bite-sized pieces and tear pita into large strips. Set aside until ready to assemble.
Whisk lemon juice, honey, zest, 2 teaspoons salt, ½ teaspoon pepper and remaining za’atar oil until completely combined.
Arrange eggplant, onion, pita, tomatoes, feta, cucumber and olives on a large platter. Drizzle dressing over top and generously coat the eggplant. Top with basil leaves and dill sprigs.