Preheat your Traeger grill to 500F (see recipe notes if using an oven). Using a fork, poke some shallow holes all over the eggplants. This will help with peeling them after cooking.
Transfer the eggplants to the back perimeter of the grill where it gets hottest. Cook, turning once or twice, until blackened all over and very soft in the middle, around 40 minutes total. Transfer the eggplants to a wooden board. Let cool for a few minutes and peel off the skin, keeping the stem intact for presentation purposes. Transfer to a serving platter and use a flat wooden spatula to roughly chop the flesh. Top with the garlic and spread as evenly as possible across the eggplant. Cover and set aside.
Bring a small pot of water to a boil. Using a slotted spoon, submerge the eggs and cook for exactly 12 minutes for hard-boiled. Transfer the eggs to a bowl of ice-cold water for 4 to 5 minutes. Peel the shells and grate them with the coarse side of a cheese grater. Set aside.
In a small bowl, combine the onion, sumac and half of the lemon juice. Toss to coat and set aside.
Shape the pizza dough into 4 equal-sized rounds and drizzle both sides with olive oil. Transfer directly onto the grill grates and cook until grill marks form, around 5 minutes. Flip and cook the other side until grill marks form, around 5 more minutes. Transfer to a plate, cover with a clean towel to keep warm and set aside.
Garnish the eggplant with the sumac onions, grated eggs, parsley, dill, and sliced chiles. Squeeze the tomatoes overtop to expel the seeds and drizzle with olive oil. Season with salt and pepper and squeeze the reserved lemon overtop. Serve with the grilled pizza bianca.