Prepare a high gas or charcoal grill fire.
In a small bowl, combine the oil with 3 Tbs. each of the thyme and scallions, the chiles, capers, lemon juice, and 1 Tbs. of the lemon zest. Season to taste with salt and pepper. (The salsa can be prepared up to 2 hours ahead; let it stand at room temperature.)
Set the fish skin side down on a small baking sheet. Arrange the eggplant rounds on a large baking sheet. Sprinkle the flesh side of the fish with salt, pepper, the remaining 2 Tbs. scallions, 2 tsp. thyme, and 2 tsp. lemon zest. Gently brush with olive oil. Brush the eggplant on both sides with olive oil and sprinkle with salt and pepper.
Grill the eggplant, covered, until tender, about 3 minutes per side. Transfer to a tray; cover and keep warm while cooking the fish.
On a gas grill, turn off one of the burners. On a charcoal grill, use a long-handled spatula to push the coals to one side of the grill. Arrange the fish skin side down on the grill grate over the cooler part of the fire, cover, and cook without turning until just cooked through, 8 to 10 minutes.
Arrange the fish and eggplant on dinner plates, spoon the salsa verde over, and serve immediately.