Preheat the oven to 450 degrees F (230 degrees C). Line a baking sheet with parchment paper or a silicone baking mat.
Combine eggplant, olive oil, salt, and black pepper in a large skillet over medium-high heat. Cook and stir until eggplant begins to soften, about 5 minutes.
Mix in 1/3 cup onion and red pepper flakes. Reduce heat to medium; cook and stir until onion is softened, about 4 minutes. Stir in broth and marinara sauce. Reduce heat to medium-low and simmer until sauce mixture has reduced by half, about 30 minutes.
Meanwhile, whisk together bread crumbs, egg, and Greek yogurt in a large bowl until bread crumbs absorb all the liquid, about 3 minutes. Mix in 1/4 cup onion, garlic, kosher salt, black pepper, cumin, cinnamon, and coriander until well combined. Crumble in lamb and sprinkle with cayenne pepper; mix well.
Form lamb mixture into small meatballs and transfer to the prepared baking sheet.
Bake in the preheated oven until browned but still pink inside, about 10 minutes.
Remove meatballs from the oven and stir into simmering sauce. Simmer over low heat until sauce begins to thicken and meatballs are cooked through, 30 to 45 minutes. An instant-read thermometer inserted into the center of a meatball should read 160 degrees F (72 degrees C).