1. Make the pickled red onion: Combine onion, lemon juice, vinegar, and sugar in a small bowl. Refrigerate for at least 4 hours or up to overnight. Drain.

2. Make the flatbreads: Heat broiler with rack 6 inches from heating element. Pierce eggplants all over with a fork. Place on a rimmed baking sheet and broil, flipping once, until charred on both sides, 20 minutes. When cool enough to handle, cut open, then scrape out flesh and discard skin. (You should have about 1 1/2 cups.) Using a fork, mash eggplant, then season with salt and stir in 2 tablespoons oil. Set aside.

3. Whisk to combine flour, baking powder, nigella seeds, and 2 teaspoons salt in a medium bowl. Add yogurt and stir until mixture comes together and forms a dough. Knead dough in bowl until soft and sticky, about 1 minute. Cover bowl with plastic wrap and let rest 15 minutes. On a lightly floured surface, divide dough into 4 equal pieces. Roll each piece into an 11-by-4-inch oval, about 1/8 inch thick.

4. Heat a large cast-iron griddle or skillet over medium-high. Brush with 1 tablespoon ghee. Working with 1 oval of dough at a time, brush with ghee, then cook flatbread ghee-side down, turning once, until deep brown, about 2 minutes per side.

5. In a small bowl, toss arugula and mint with remaining 2 tablespoons oil; season with salt.

6. Make the tahina sauce: Mix tahini, lemon juice, oil, paprika, and cayenne in a small bowl until smooth; season with salt.

7. To serve, divide the eggplant mixture among the 4 flatbreads and spread evenly to cover. Top each with arugula mixture and pickled onion. Cut each flatbread crosswise into 8 triangles, drizzle with tahina sauce, and serve.